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) 1999-2024 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{https://www.ginac.de}
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2024 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{https://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{} 1999-2024 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, factorized, and normalized (i.e. converted to a ratio of
376 two coprime polynomials):
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*y+x^2-y^2)^2*(x^2+3*y^2)
388 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
390 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
395 Here we have made use of the @command{ginsh}-command @code{%} to pop the
396 previously evaluated element from @command{ginsh}'s internal stack.
398 You can differentiate functions and expand them as Taylor or Laurent
399 series in a very natural syntax (the second argument of @code{series} is
400 a relation defining the evaluation point, the third specifies the
403 @cindex Zeta function
407 > series(sin(x),x==0,4);
409 > series(1/tan(x),x==0,4);
410 x^(-1)-1/3*x+Order(x^2)
411 > series(tgamma(x),x==0,3);
412 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
413 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
415 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
416 -(0.90747907608088628905)*x^2+Order(x^3)
417 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
418 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
419 -Euler-1/12+Order((x-1/2*Pi)^3)
422 Often, functions don't have roots in closed form. Nevertheless, it's
423 quite easy to compute a solution numerically, to arbitrary precision:
428 > fsolve(cos(x)==x,x,0,2);
429 0.7390851332151606416553120876738734040134117589007574649658
431 > X=fsolve(f,x,-10,10);
432 2.2191071489137460325957851882042901681753665565320678854155
434 -6.372367644529809108115521591070847222364418220770475144296E-58
437 Notice how the final result above differs slightly from zero by about
438 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
439 root cannot be represented more accurately than @code{X}. Such
440 inaccuracies are to be expected when computing with finite floating
443 If you ever wanted to convert units in C or C++ and found this is
444 cumbersome, here is the solution. Symbolic types can always be used as
445 tags for different types of objects. Converting from wrong units to the
446 metric system is now easy:
454 140613.91592783185568*kg*m^(-2)
458 @node Installation, Prerequisites, What it can do for you, Top
459 @c node-name, next, previous, up
460 @chapter Installation
463 GiNaC's installation follows the spirit of most GNU software. It is
464 easily installed on your system by three steps: configuration, build,
468 * Prerequisites:: Packages upon which GiNaC depends.
469 * Configuration:: How to configure GiNaC.
470 * Building GiNaC:: How to compile GiNaC.
471 * Installing GiNaC:: How to install GiNaC on your system.
475 @node Prerequisites, Configuration, Installation, Installation
476 @c node-name, next, previous, up
477 @section Prerequisites
479 In order to install GiNaC on your system, some prerequisites need to be
480 met. First of all, you need to have a C++-compiler adhering to the
481 ISO standard @cite{ISO/IEC 14882:2011(E)}. We used GCC for development
482 so if you have a different compiler you are on your own. For the
483 configuration to succeed you need a Posix compliant shell installed in
484 @file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
485 required for the configuration, it can be downloaded from
486 @uref{http://pkg-config.freedesktop.org}.
487 Last but not least, the CLN library
488 is used extensively and needs to be installed on your system.
489 Please get it from @uref{https://www.ginac.de/CLN/} (it is licensed under
490 the GPL) and install it prior to trying to install GiNaC. The configure
491 script checks if it can find it and if it cannot, it will refuse to
495 @node Configuration, Building GiNaC, Prerequisites, Installation
496 @c node-name, next, previous, up
497 @section Configuration
498 @cindex configuration
501 To configure GiNaC means to prepare the source distribution for
502 building. It is done via a shell script called @command{configure} that
503 is shipped with the sources and was originally generated by GNU
504 Autoconf. Since a configure script generated by GNU Autoconf never
505 prompts, all customization must be done either via command line
506 parameters or environment variables. It accepts a list of parameters,
507 the complete set of which can be listed by calling it with the
508 @option{--help} option. The most important ones will be shortly
509 described in what follows:
514 @option{--disable-shared}: When given, this option switches off the
515 build of a shared library, i.e. a @file{.so} file. This may be convenient
516 when developing because it considerably speeds up compilation.
519 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
520 and headers are installed. It defaults to @file{/usr/local} which means
521 that the library is installed in the directory @file{/usr/local/lib},
522 the header files in @file{/usr/local/include/ginac} and the documentation
523 (like this one) into @file{/usr/local/share/doc/GiNaC}.
526 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
527 the library installed in some other directory than
528 @file{@var{PREFIX}/lib/}.
531 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
532 to have the header files installed in some other directory than
533 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
534 @option{--includedir=/usr/include} you will end up with the header files
535 sitting in the directory @file{/usr/include/ginac/}. Note that the
536 subdirectory @file{ginac} is enforced by this process in order to
537 keep the header files separated from others. This avoids some
538 clashes and allows for an easier deinstallation of GiNaC. This ought
539 to be considered A Good Thing (tm).
542 @option{--datadir=@var{DATADIR}}: This option may be given in case you
543 want to have the documentation installed in some other directory than
544 @file{@var{PREFIX}/share/doc/GiNaC/}.
548 In addition, you may specify some environment variables. @env{CXX}
549 holds the path and the name of the C++ compiler in case you want to
550 override the default in your path. (The @command{configure} script
551 searches your path for @command{c++}, @command{g++}, @command{gcc},
552 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
553 be very useful to define some compiler flags with the @env{CXXFLAGS}
554 environment variable, like optimization, debugging information and
555 warning levels. If omitted, it defaults to @option{-g
556 -O2}.@footnote{The @command{configure} script is itself generated from
557 the file @file{configure.ac}. It is only distributed in packaged
558 releases of GiNaC. If you got the naked sources, e.g. from git, you
559 must generate @command{configure} along with the various
560 @file{Makefile.in} by using the @command{autoreconf} utility. This will
561 require a fair amount of support from your local toolchain, though.}
563 The whole process is illustrated in the following two
564 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
565 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
568 Here is a simple configuration for a site-wide GiNaC library assuming
569 everything is in default paths:
572 $ export CXXFLAGS="-Wall -O2"
576 And here is a configuration for a private static GiNaC library with
577 several components sitting in custom places (site-wide GCC and private
578 CLN). The compiler is persuaded to be picky and full assertions and
579 debugging information are switched on:
582 $ export CXX=/usr/local/gnu/bin/c++
583 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
584 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
585 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
586 $ ./configure --disable-shared --prefix=$(HOME)
590 @node Building GiNaC, Installing GiNaC, Configuration, Installation
591 @c node-name, next, previous, up
592 @section Building GiNaC
593 @cindex building GiNaC
595 After proper configuration you should just build the whole
600 at the command prompt and go for a cup of coffee. The exact time it
601 takes to compile GiNaC depends not only on the speed of your machines
602 but also on other parameters, for instance what value for @env{CXXFLAGS}
603 you entered. Optimization may be very time-consuming.
605 Just to make sure GiNaC works properly you may run a collection of
606 regression tests by typing
612 This will compile some sample programs, run them and check the output
613 for correctness. The regression tests fall in three categories. First,
614 the so called @emph{exams} are performed, simple tests where some
615 predefined input is evaluated (like a pupils' exam). Second, the
616 @emph{checks} test the coherence of results among each other with
617 possible random input. Third, some @emph{timings} are performed, which
618 benchmark some predefined problems with different sizes and display the
619 CPU time used in seconds. Each individual test should return a message
620 @samp{passed}. This is mostly intended to be a QA-check if something
621 was broken during development, not a sanity check of your system. Some
622 of the tests in sections @emph{checks} and @emph{timings} may require
623 insane amounts of memory and CPU time. Feel free to kill them if your
624 machine catches fire. Another quite important intent is to allow people
625 to fiddle around with optimization.
627 By default, the only documentation that will be built is this tutorial
628 in @file{.info} format. To build the GiNaC tutorial and reference manual
629 in HTML, DVI, PostScript, or PDF formats, use one of
638 Generally, the top-level Makefile runs recursively to the
639 subdirectories. It is therefore safe to go into any subdirectory
640 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
641 @var{target} there in case something went wrong.
644 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
645 @c node-name, next, previous, up
646 @section Installing GiNaC
649 To install GiNaC on your system, simply type
655 As described in the section about configuration the files will be
656 installed in the following directories (the directories will be created
657 if they don't already exist):
662 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
663 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
664 So will @file{libginac.so} unless the configure script was
665 given the option @option{--disable-shared}. The proper symlinks
666 will be established as well.
669 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
670 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
673 All documentation (info) will be stuffed into
674 @file{@var{PREFIX}/share/doc/GiNaC/} (or
675 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
679 For the sake of completeness we will list some other useful make
680 targets: @command{make clean} deletes all files generated by
681 @command{make}, i.e. all the object files. In addition @command{make
682 distclean} removes all files generated by the configuration and
683 @command{make maintainer-clean} goes one step further and deletes files
684 that may require special tools to rebuild (like the @command{libtool}
685 for instance). Finally @command{make uninstall} removes the installed
686 library, header files and documentation@footnote{Uninstallation does not
687 work after you have called @command{make distclean} since the
688 @file{Makefile} is itself generated by the configuration from
689 @file{Makefile.in} and hence deleted by @command{make distclean}. There
690 are two obvious ways out of this dilemma. First, you can run the
691 configuration again with the same @var{PREFIX} thus creating a
692 @file{Makefile} with a working @samp{uninstall} target. Second, you can
693 do it by hand since you now know where all the files went during
697 @node Basic concepts, Expressions, Installing GiNaC, Top
698 @c node-name, next, previous, up
699 @chapter Basic concepts
701 This chapter will describe the different fundamental objects that can be
702 handled by GiNaC. But before doing so, it is worthwhile introducing you
703 to the more commonly used class of expressions, representing a flexible
704 meta-class for storing all mathematical objects.
707 * Expressions:: The fundamental GiNaC class.
708 * Automatic evaluation:: Evaluation and canonicalization.
709 * Error handling:: How the library reports errors.
710 * The class hierarchy:: Overview of GiNaC's classes.
711 * Symbols:: Symbolic objects.
712 * Numbers:: Numerical objects.
713 * Constants:: Pre-defined constants.
714 * Fundamental containers:: Sums, products and powers.
715 * Lists:: Lists of expressions.
716 * Mathematical functions:: Mathematical functions.
717 * Relations:: Equality, Inequality and all that.
718 * Integrals:: Symbolic integrals.
719 * Matrices:: Matrices.
720 * Indexed objects:: Handling indexed quantities.
721 * Non-commutative objects:: Algebras with non-commutative products.
725 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
726 @c node-name, next, previous, up
728 @cindex expression (class @code{ex})
731 The most common class of objects a user deals with is the expression
732 @code{ex}, representing a mathematical object like a variable, number,
733 function, sum, product, etc@dots{} Expressions may be put together to form
734 new expressions, passed as arguments to functions, and so on. Here is a
735 little collection of valid expressions:
738 ex MyEx1 = 5; // simple number
739 ex MyEx2 = x + 2*y; // polynomial in x and y
740 ex MyEx3 = (x + 1)/(x - 1); // rational expression
741 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
742 ex MyEx5 = MyEx4 + 1; // similar to above
745 Expressions are handles to other more fundamental objects, that often
746 contain other expressions thus creating a tree of expressions
747 (@xref{Internal structures}, for particular examples). Most methods on
748 @code{ex} therefore run top-down through such an expression tree. For
749 example, the method @code{has()} scans recursively for occurrences of
750 something inside an expression. Thus, if you have declared @code{MyEx4}
751 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
752 the argument of @code{sin} and hence return @code{true}.
754 The next sections will outline the general picture of GiNaC's class
755 hierarchy and describe the classes of objects that are handled by
758 @subsection Note: Expressions and STL containers
760 GiNaC expressions (@code{ex} objects) have value semantics (they can be
761 assigned, reassigned and copied like integral types) but the operator
762 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
763 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
765 This implies that in order to use expressions in sorted containers such as
766 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
767 comparison predicate. GiNaC provides such a predicate, called
768 @code{ex_is_less}. For example, a set of expressions should be defined
769 as @code{std::set<ex, ex_is_less>}.
771 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
772 don't pose a problem. A @code{std::vector<ex>} works as expected.
774 @xref{Information about expressions}, for more about comparing and ordering
778 @node Automatic evaluation, Error handling, Expressions, Basic concepts
779 @c node-name, next, previous, up
780 @section Automatic evaluation and canonicalization of expressions
783 GiNaC performs some automatic transformations on expressions, to simplify
784 them and put them into a canonical form. Some examples:
787 ex MyEx1 = 2*x - 1 + x; // 3*x-1
788 ex MyEx2 = x - x; // 0
789 ex MyEx3 = cos(2*Pi); // 1
790 ex MyEx4 = x*y/x; // y
793 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
794 evaluation}. GiNaC only performs transformations that are
798 at most of complexity
806 algebraically correct, possibly except for a set of measure zero (e.g.
807 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
810 There are two types of automatic transformations in GiNaC that may not
811 behave in an entirely obvious way at first glance:
815 The terms of sums and products (and some other things like the arguments of
816 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
817 into a canonical form that is deterministic, but not lexicographical or in
818 any other way easy to guess (it almost always depends on the number and
819 order of the symbols you define). However, constructing the same expression
820 twice, either implicitly or explicitly, will always result in the same
823 Expressions of the form 'number times sum' are automatically expanded (this
824 has to do with GiNaC's internal representation of sums and products). For
827 ex MyEx5 = 2*(x + y); // 2*x+2*y
828 ex MyEx6 = z*(x + y); // z*(x+y)
832 The general rule is that when you construct expressions, GiNaC automatically
833 creates them in canonical form, which might differ from the form you typed in
834 your program. This may create some awkward looking output (@samp{-y+x} instead
835 of @samp{x-y}) but allows for more efficient operation and usually yields
836 some immediate simplifications.
838 @cindex @code{eval()}
839 Internally, the anonymous evaluator in GiNaC is implemented by the methods
843 ex basic::eval() const;
846 but unless you are extending GiNaC with your own classes or functions, there
847 should never be any reason to call them explicitly. All GiNaC methods that
848 transform expressions, like @code{subs()} or @code{normal()}, automatically
849 re-evaluate their results.
852 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
853 @c node-name, next, previous, up
854 @section Error handling
856 @cindex @code{pole_error} (class)
858 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
859 generated by GiNaC are subclassed from the standard @code{exception} class
860 defined in the @file{<stdexcept>} header. In addition to the predefined
861 @code{logic_error}, @code{domain_error}, @code{out_of_range},
862 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
863 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
864 exception that gets thrown when trying to evaluate a mathematical function
867 The @code{pole_error} class has a member function
870 int pole_error::degree() const;
873 that returns the order of the singularity (or 0 when the pole is
874 logarithmic or the order is undefined).
876 When using GiNaC it is useful to arrange for exceptions to be caught in
877 the main program even if you don't want to do any special error handling.
878 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
879 default exception handler of your C++ compiler's run-time system which
880 usually only aborts the program without giving any information what went
883 Here is an example for a @code{main()} function that catches and prints
884 exceptions generated by GiNaC:
889 #include <ginac/ginac.h>
891 using namespace GiNaC;
899 @} catch (exception &p) @{
900 cerr << p.what() << endl;
908 @node The class hierarchy, Symbols, Error handling, Basic concepts
909 @c node-name, next, previous, up
910 @section The class hierarchy
912 GiNaC's class hierarchy consists of several classes representing
913 mathematical objects, all of which (except for @code{ex} and some
914 helpers) are internally derived from one abstract base class called
915 @code{basic}. You do not have to deal with objects of class
916 @code{basic}, instead you'll be dealing with symbols, numbers,
917 containers of expressions and so on.
921 To get an idea about what kinds of symbolic composites may be built we
922 have a look at the most important classes in the class hierarchy and
923 some of the relations among the classes:
926 @image{classhierarchy}
932 The abstract classes shown here (the ones without drop-shadow) are of no
933 interest for the user. They are used internally in order to avoid code
934 duplication if two or more classes derived from them share certain
935 features. An example is @code{expairseq}, a container for a sequence of
936 pairs each consisting of one expression and a number (@code{numeric}).
937 What @emph{is} visible to the user are the derived classes @code{add}
938 and @code{mul}, representing sums and products. @xref{Internal
939 structures}, where these two classes are described in more detail. The
940 following table shortly summarizes what kinds of mathematical objects
941 are stored in the different classes:
944 @multitable @columnfractions .22 .78
945 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
946 @item @code{constant} @tab Constants like
953 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
954 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
955 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
956 @item @code{ncmul} @tab Products of non-commutative objects
957 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
962 @code{sqrt(}@math{2}@code{)}
965 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
966 @item @code{function} @tab A symbolic function like
973 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
974 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
975 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
976 @item @code{indexed} @tab Indexed object like @math{A_ij}
977 @item @code{tensor} @tab Special tensor like the delta and metric tensors
978 @item @code{idx} @tab Index of an indexed object
979 @item @code{varidx} @tab Index with variance
980 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
981 @item @code{wildcard} @tab Wildcard for pattern matching
982 @item @code{structure} @tab Template for user-defined classes
987 @node Symbols, Numbers, The class hierarchy, Basic concepts
988 @c node-name, next, previous, up
990 @cindex @code{symbol} (class)
991 @cindex hierarchy of classes
994 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
995 manipulation what atoms are for chemistry.
997 A typical symbol definition looks like this:
1002 This definition actually contains three very different things:
1004 @item a C++ variable named @code{x}
1005 @item a @code{symbol} object stored in this C++ variable; this object
1006 represents the symbol in a GiNaC expression
1007 @item the string @code{"x"} which is the name of the symbol, used (almost)
1008 exclusively for printing expressions holding the symbol
1011 Symbols have an explicit name, supplied as a string during construction,
1012 because in C++, variable names can't be used as values, and the C++ compiler
1013 throws them away during compilation.
1015 It is possible to omit the symbol name in the definition:
1020 In this case, GiNaC will assign the symbol an internal, unique name of the
1021 form @code{symbolNNN}. This won't affect the usability of the symbol but
1022 the output of your calculations will become more readable if you give your
1023 symbols sensible names (for intermediate expressions that are only used
1024 internally such anonymous symbols can be quite useful, however).
1026 Now, here is one important property of GiNaC that differentiates it from
1027 other computer algebra programs you may have used: GiNaC does @emph{not} use
1028 the names of symbols to tell them apart, but a (hidden) serial number that
1029 is unique for each newly created @code{symbol} object. If you want to use
1030 one and the same symbol in different places in your program, you must only
1031 create one @code{symbol} object and pass that around. If you create another
1032 symbol, even if it has the same name, GiNaC will treat it as a different
1049 // prints "x^6" which looks right, but...
1051 cout << e.degree(x) << endl;
1052 // ...this doesn't work. The symbol "x" here is different from the one
1053 // in f() and in the expression returned by f(). Consequently, it
1058 One possibility to ensure that @code{f()} and @code{main()} use the same
1059 symbol is to pass the symbol as an argument to @code{f()}:
1061 ex f(int n, const ex & x)
1070 // Now, f() uses the same symbol.
1073 cout << e.degree(x) << endl;
1074 // prints "6", as expected
1078 Another possibility would be to define a global symbol @code{x} that is used
1079 by both @code{f()} and @code{main()}. If you are using global symbols and
1080 multiple compilation units you must take special care, however. Suppose
1081 that you have a header file @file{globals.h} in your program that defines
1082 a @code{symbol x("x");}. In this case, every unit that includes
1083 @file{globals.h} would also get its own definition of @code{x} (because
1084 header files are just inlined into the source code by the C++ preprocessor),
1085 and hence you would again end up with multiple equally-named, but different,
1086 symbols. Instead, the @file{globals.h} header should only contain a
1087 @emph{declaration} like @code{extern symbol x;}, with the definition of
1088 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1090 A different approach to ensuring that symbols used in different parts of
1091 your program are identical is to create them with a @emph{factory} function
1094 const symbol & get_symbol(const string & s)
1096 static map<string, symbol> directory;
1097 map<string, symbol>::iterator i = directory.find(s);
1098 if (i != directory.end())
1101 return directory.insert(make_pair(s, symbol(s))).first->second;
1105 This function returns one newly constructed symbol for each name that is
1106 passed in, and it returns the same symbol when called multiple times with
1107 the same name. Using this symbol factory, we can rewrite our example like
1112 return pow(get_symbol("x"), n);
1119 // Both calls of get_symbol("x") yield the same symbol.
1120 cout << e.degree(get_symbol("x")) << endl;
1125 Instead of creating symbols from strings we could also have
1126 @code{get_symbol()} take, for example, an integer number as its argument.
1127 In this case, we would probably want to give the generated symbols names
1128 that include this number, which can be accomplished with the help of an
1129 @code{ostringstream}.
1131 In general, if you're getting weird results from GiNaC such as an expression
1132 @samp{x-x} that is not simplified to zero, you should check your symbol
1135 As we said, the names of symbols primarily serve for purposes of expression
1136 output. But there are actually two instances where GiNaC uses the names for
1137 identifying symbols: When constructing an expression from a string, and when
1138 recreating an expression from an archive (@pxref{Input/output}).
1140 In addition to its name, a symbol may contain a special string that is used
1143 symbol x("x", "\\Box");
1146 This creates a symbol that is printed as "@code{x}" in normal output, but
1147 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1148 information about the different output formats of expressions in GiNaC).
1149 GiNaC automatically creates proper LaTeX code for symbols having names of
1150 greek letters (@samp{alpha}, @samp{mu}, etc.). You can retrieve the name
1151 and the LaTeX name of a symbol using the respective methods:
1152 @cindex @code{get_name()}
1153 @cindex @code{get_TeX_name()}
1155 symbol::get_name() const;
1156 symbol::get_TeX_name() const;
1159 @cindex @code{subs()}
1160 Symbols in GiNaC can't be assigned values. If you need to store results of
1161 calculations and give them a name, use C++ variables of type @code{ex}.
1162 If you want to replace a symbol in an expression with something else, you
1163 can invoke the expression's @code{.subs()} method
1164 (@pxref{Substituting expressions}).
1166 @cindex @code{realsymbol()}
1167 By default, symbols are expected to stand in for complex values, i.e. they live
1168 in the complex domain. As a consequence, operations like complex conjugation,
1169 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1170 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1171 because of the unknown imaginary part of @code{x}.
1172 On the other hand, if you are sure that your symbols will hold only real
1173 values, you would like to have such functions evaluated. Therefore GiNaC
1174 allows you to specify
1175 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1176 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1178 @cindex @code{possymbol()}
1179 Furthermore, it is also possible to declare a symbol as positive. This will,
1180 for instance, enable the automatic simplification of @code{abs(x)} into
1181 @code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.
1184 @node Numbers, Constants, Symbols, Basic concepts
1185 @c node-name, next, previous, up
1187 @cindex @code{numeric} (class)
1193 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1194 The classes therein serve as foundation classes for GiNaC. CLN stands
1195 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1196 In order to find out more about CLN's internals, the reader is referred to
1197 the documentation of that library. @xref{Top,,, cln, The CLN Manual}, for
1198 more information. Suffice to say that it is by itself build on top of
1199 another library, the GNU Multiple Precision library GMP, which is an
1200 extremely fast library for arbitrary long integers and rationals as well
1201 as arbitrary precision floating point numbers. It is very commonly used
1202 by several popular cryptographic applications. CLN extends GMP by
1203 several useful things: First, it introduces the complex number field
1204 over either reals (i.e. floating point numbers with arbitrary precision)
1205 or rationals. Second, it automatically converts rationals to integers
1206 if the denominator is unity and complex numbers to real numbers if the
1207 imaginary part vanishes and also correctly treats algebraic functions.
1208 Third it provides good implementations of state-of-the-art algorithms
1209 for all trigonometric and hyperbolic functions as well as for
1210 calculation of some useful constants.
1212 The user can construct an object of class @code{numeric} in several
1213 ways. The following example shows the four most important constructors.
1214 It uses construction from C-integer, construction of fractions from two
1215 integers, construction from C-float and construction from a string:
1219 #include <ginac/ginac.h>
1220 using namespace GiNaC;
1224 numeric two = 2; // exact integer 2
1225 numeric r(2,3); // exact fraction 2/3
1226 numeric e(2.71828); // floating point number
1227 numeric p = "3.14159265358979323846"; // constructor from string
1228 // Trott's constant in scientific notation:
1229 numeric trott("1.0841015122311136151E-2");
1231 std::cout << two*p << std::endl; // floating point 6.283...
1236 @cindex complex numbers
1237 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1242 numeric z1 = 2-3*I; // exact complex number 2-3i
1243 numeric z2 = 5.9+1.6*I; // complex floating point number
1247 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1248 This would, however, call C's built-in operator @code{/} for integers
1249 first and result in a numeric holding a plain integer 1. @strong{Never
1250 use the operator @code{/} on integers} unless you know exactly what you
1251 are doing! Use the constructor from two integers instead, as shown in
1252 the example above. Writing @code{numeric(1)/2} may look funny but works
1255 @cindex @code{Digits}
1257 We have seen now the distinction between exact numbers and floating
1258 point numbers. Clearly, the user should never have to worry about
1259 dynamically created exact numbers, since their `exactness' always
1260 determines how they ought to be handled, i.e. how `long' they are. The
1261 situation is different for floating point numbers. Their accuracy is
1262 controlled by one @emph{global} variable, called @code{Digits}. (For
1263 those readers who know about Maple: it behaves very much like Maple's
1264 @code{Digits}). All objects of class numeric that are constructed from
1265 then on will be stored with a precision matching that number of decimal
1270 #include <ginac/ginac.h>
1271 using namespace std;
1272 using namespace GiNaC;
1276 numeric three(3.0), one(1.0);
1277 numeric x = one/three;
1279 cout << "in " << Digits << " digits:" << endl;
1281 cout << Pi.evalf() << endl;
1293 The above example prints the following output to screen:
1297 0.33333333333333333334
1298 3.1415926535897932385
1300 0.33333333333333333333333333333333333333333333333333333333333333333334
1301 3.1415926535897932384626433832795028841971693993751058209749445923078
1305 Note that the last number is not necessarily rounded as you would
1306 naively expect it to be rounded in the decimal system. But note also,
1307 that in both cases you got a couple of extra digits. This is because
1308 numbers are internally stored by CLN as chunks of binary digits in order
1309 to match your machine's word size and to not waste precision. Thus, on
1310 architectures with different word size, the above output might even
1311 differ with regard to actually computed digits.
1313 It should be clear that objects of class @code{numeric} should be used
1314 for constructing numbers or for doing arithmetic with them. The objects
1315 one deals with most of the time are the polymorphic expressions @code{ex}.
1317 @subsection Tests on numbers
1319 Once you have declared some numbers, assigned them to expressions and
1320 done some arithmetic with them it is frequently desired to retrieve some
1321 kind of information from them like asking whether that number is
1322 integer, rational, real or complex. For those cases GiNaC provides
1323 several useful methods. (Internally, they fall back to invocations of
1324 certain CLN functions.)
1326 As an example, let's construct some rational number, multiply it with
1327 some multiple of its denominator and test what comes out:
1331 #include <ginac/ginac.h>
1332 using namespace std;
1333 using namespace GiNaC;
1335 // some very important constants:
1336 const numeric twentyone(21);
1337 const numeric ten(10);
1338 const numeric five(5);
1342 numeric answer = twentyone;
1345 cout << answer.is_integer() << endl; // false, it's 21/5
1347 cout << answer.is_integer() << endl; // true, it's 42 now!
1351 Note that the variable @code{answer} is constructed here as an integer
1352 by @code{numeric}'s copy constructor, but in an intermediate step it
1353 holds a rational number represented as integer numerator and integer
1354 denominator. When multiplied by 10, the denominator becomes unity and
1355 the result is automatically converted to a pure integer again.
1356 Internally, the underlying CLN is responsible for this behavior and we
1357 refer the reader to CLN's documentation. Suffice to say that
1358 the same behavior applies to complex numbers as well as return values of
1359 certain functions. Complex numbers are automatically converted to real
1360 numbers if the imaginary part becomes zero. The full set of tests that
1361 can be applied is listed in the following table.
1364 @multitable @columnfractions .30 .70
1365 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1366 @item @code{.is_zero()}
1367 @tab @dots{}equal to zero
1368 @item @code{.is_positive()}
1369 @tab @dots{}not complex and greater than 0
1370 @item @code{.is_negative()}
1371 @tab @dots{}not complex and smaller than 0
1372 @item @code{.is_integer()}
1373 @tab @dots{}a (non-complex) integer
1374 @item @code{.is_pos_integer()}
1375 @tab @dots{}an integer and greater than 0
1376 @item @code{.is_nonneg_integer()}
1377 @tab @dots{}an integer and greater equal 0
1378 @item @code{.is_even()}
1379 @tab @dots{}an even integer
1380 @item @code{.is_odd()}
1381 @tab @dots{}an odd integer
1382 @item @code{.is_prime()}
1383 @tab @dots{}a prime integer (probabilistic primality test)
1384 @item @code{.is_rational()}
1385 @tab @dots{}an exact rational number (integers are rational, too)
1386 @item @code{.is_real()}
1387 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1388 @item @code{.is_cinteger()}
1389 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1390 @item @code{.is_crational()}
1391 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1397 @subsection Numeric functions
1399 The following functions can be applied to @code{numeric} objects and will be
1400 evaluated immediately:
1403 @multitable @columnfractions .30 .70
1404 @item @strong{Name} @tab @strong{Function}
1405 @item @code{inverse(z)}
1406 @tab returns @math{1/z}
1407 @cindex @code{inverse()} (numeric)
1408 @item @code{pow(a, b)}
1409 @tab exponentiation @math{a^b}
1412 @item @code{real(z)}
1414 @cindex @code{real()}
1415 @item @code{imag(z)}
1417 @cindex @code{imag()}
1418 @item @code{csgn(z)}
1419 @tab complex sign (returns an @code{int})
1420 @item @code{step(x)}
1421 @tab step function (returns an @code{numeric})
1422 @item @code{numer(z)}
1423 @tab numerator of rational or complex rational number
1424 @item @code{denom(z)}
1425 @tab denominator of rational or complex rational number
1426 @item @code{sqrt(z)}
1428 @item @code{isqrt(n)}
1429 @tab integer square root
1430 @cindex @code{isqrt()}
1437 @item @code{asin(z)}
1439 @item @code{acos(z)}
1441 @item @code{atan(z)}
1442 @tab inverse tangent
1443 @item @code{atan(y, x)}
1444 @tab inverse tangent with two arguments
1445 @item @code{sinh(z)}
1446 @tab hyperbolic sine
1447 @item @code{cosh(z)}
1448 @tab hyperbolic cosine
1449 @item @code{tanh(z)}
1450 @tab hyperbolic tangent
1451 @item @code{asinh(z)}
1452 @tab inverse hyperbolic sine
1453 @item @code{acosh(z)}
1454 @tab inverse hyperbolic cosine
1455 @item @code{atanh(z)}
1456 @tab inverse hyperbolic tangent
1458 @tab exponential function
1460 @tab natural logarithm
1463 @item @code{zeta(z)}
1464 @tab Riemann's zeta function
1465 @item @code{tgamma(z)}
1467 @item @code{lgamma(z)}
1468 @tab logarithm of gamma function
1470 @tab psi (digamma) function
1471 @item @code{psi(n, z)}
1472 @tab derivatives of psi function (polygamma functions)
1473 @item @code{factorial(n)}
1474 @tab factorial function @math{n!}
1475 @item @code{doublefactorial(n)}
1476 @tab double factorial function @math{n!!}
1477 @cindex @code{doublefactorial()}
1478 @item @code{binomial(n, k)}
1479 @tab binomial coefficients
1480 @item @code{bernoulli(n)}
1481 @tab Bernoulli numbers
1482 @cindex @code{bernoulli()}
1483 @item @code{fibonacci(n)}
1484 @tab Fibonacci numbers
1485 @cindex @code{fibonacci()}
1486 @item @code{mod(a, b)}
1487 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1488 @cindex @code{mod()}
1489 @item @code{smod(a, b)}
1490 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
1491 @cindex @code{smod()}
1492 @item @code{irem(a, b)}
1493 @tab integer remainder (has the sign of @math{a}, or is zero)
1494 @cindex @code{irem()}
1495 @item @code{irem(a, b, q)}
1496 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1497 @item @code{iquo(a, b)}
1498 @tab integer quotient
1499 @cindex @code{iquo()}
1500 @item @code{iquo(a, b, r)}
1501 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1502 @item @code{gcd(a, b)}
1503 @tab greatest common divisor
1504 @item @code{lcm(a, b)}
1505 @tab least common multiple
1509 Most of these functions are also available as symbolic functions that can be
1510 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1511 as polynomial algorithms.
1513 @subsection Converting numbers
1515 Sometimes it is desirable to convert a @code{numeric} object back to a
1516 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1517 class provides a couple of methods for this purpose:
1519 @cindex @code{to_int()}
1520 @cindex @code{to_long()}
1521 @cindex @code{to_double()}
1522 @cindex @code{to_cl_N()}
1524 int numeric::to_int() const;
1525 long numeric::to_long() const;
1526 double numeric::to_double() const;
1527 cln::cl_N numeric::to_cl_N() const;
1530 @code{to_int()} and @code{to_long()} only work when the number they are
1531 applied on is an exact integer. Otherwise the program will halt with a
1532 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1533 rational number will return a floating-point approximation. Both
1534 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1535 part of complex numbers.
1537 Note the signature of the above methods, you may need to apply a type
1538 conversion and call @code{evalf()} as shown in the following example:
1541 ex e1 = 1, e2 = sin(Pi/5);
1542 cout << ex_to<numeric>(e1).to_int() << endl
1543 << ex_to<numeric>(e2.evalf()).to_double() << endl;
1547 @node Constants, Fundamental containers, Numbers, Basic concepts
1548 @c node-name, next, previous, up
1550 @cindex @code{constant} (class)
1553 @cindex @code{Catalan}
1554 @cindex @code{Euler}
1555 @cindex @code{evalf()}
1556 Constants behave pretty much like symbols except that they return some
1557 specific number when the method @code{.evalf()} is called.
1559 The predefined known constants are:
1562 @multitable @columnfractions .14 .32 .54
1563 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1565 @tab Archimedes' constant
1566 @tab 3.14159265358979323846264338327950288
1567 @item @code{Catalan}
1568 @tab Catalan's constant
1569 @tab 0.91596559417721901505460351493238411
1571 @tab Euler's (or Euler-Mascheroni) constant
1572 @tab 0.57721566490153286060651209008240243
1577 @node Fundamental containers, Lists, Constants, Basic concepts
1578 @c node-name, next, previous, up
1579 @section Sums, products and powers
1583 @cindex @code{power}
1585 Simple rational expressions are written down in GiNaC pretty much like
1586 in other CAS or like expressions involving numerical variables in C.
1587 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1588 been overloaded to achieve this goal. When you run the following
1589 code snippet, the constructor for an object of type @code{mul} is
1590 automatically called to hold the product of @code{a} and @code{b} and
1591 then the constructor for an object of type @code{add} is called to hold
1592 the sum of that @code{mul} object and the number one:
1596 symbol a("a"), b("b");
1601 @cindex @code{pow()}
1602 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1603 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1604 construction is necessary since we cannot safely overload the constructor
1605 @code{^} in C++ to construct a @code{power} object. If we did, it would
1606 have several counterintuitive and undesired effects:
1610 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1612 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1613 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1614 interpret this as @code{x^(a^b)}.
1616 Also, expressions involving integer exponents are very frequently used,
1617 which makes it even more dangerous to overload @code{^} since it is then
1618 hard to distinguish between the semantics as exponentiation and the one
1619 for exclusive or. (It would be embarrassing to return @code{1} where one
1620 has requested @code{2^3}.)
1623 @cindex @command{ginsh}
1624 All effects are contrary to mathematical notation and differ from the
1625 way most other CAS handle exponentiation, therefore overloading @code{^}
1626 is ruled out for GiNaC's C++ part. The situation is different in
1627 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1628 that the other frequently used exponentiation operator @code{**} does
1629 not exist at all in C++).
1631 To be somewhat more precise, objects of the three classes described
1632 here, are all containers for other expressions. An object of class
1633 @code{power} is best viewed as a container with two slots, one for the
1634 basis, one for the exponent. All valid GiNaC expressions can be
1635 inserted. However, basic transformations like simplifying
1636 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1637 when this is mathematically possible. If we replace the outer exponent
1638 three in the example by some symbols @code{a}, the simplification is not
1639 safe and will not be performed, since @code{a} might be @code{1/2} and
1642 Objects of type @code{add} and @code{mul} are containers with an
1643 arbitrary number of slots for expressions to be inserted. Again, simple
1644 and safe simplifications are carried out like transforming
1645 @code{3*x+4-x} to @code{2*x+4}.
1648 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1649 @c node-name, next, previous, up
1650 @section Lists of expressions
1651 @cindex @code{lst} (class)
1653 @cindex @code{nops()}
1655 @cindex @code{append()}
1656 @cindex @code{prepend()}
1657 @cindex @code{remove_first()}
1658 @cindex @code{remove_last()}
1659 @cindex @code{remove_all()}
1661 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1662 expressions. They are not as ubiquitous as in many other computer algebra
1663 packages, but are sometimes used to supply a variable number of arguments of
1664 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1665 constructors, so you should have a basic understanding of them.
1667 Lists can be constructed from an initializer list of expressions:
1671 symbol x("x"), y("y");
1672 lst l = @{x, 2, y, x+y@};
1673 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1678 Use the @code{nops()} method to determine the size (number of expressions) of
1679 a list and the @code{op()} method or the @code{[]} operator to access
1680 individual elements:
1684 cout << l.nops() << endl; // prints '4'
1685 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1689 As with the standard @code{list<T>} container, accessing random elements of a
1690 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1691 sequential access to the elements of a list is possible with the
1692 iterator types provided by the @code{lst} class:
1695 typedef ... lst::const_iterator;
1696 typedef ... lst::const_reverse_iterator;
1697 lst::const_iterator lst::begin() const;
1698 lst::const_iterator lst::end() const;
1699 lst::const_reverse_iterator lst::rbegin() const;
1700 lst::const_reverse_iterator lst::rend() const;
1703 For example, to print the elements of a list individually you can use:
1708 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1713 which is one order faster than
1718 for (size_t i = 0; i < l.nops(); ++i)
1719 cout << l.op(i) << endl;
1723 These iterators also allow you to use some of the algorithms provided by
1724 the C++ standard library:
1728 // print the elements of the list (requires #include <iterator>)
1729 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1731 // sum up the elements of the list (requires #include <numeric>)
1732 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1733 cout << sum << endl; // prints '2+2*x+2*y'
1737 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1738 (the only other one is @code{matrix}). You can modify single elements:
1742 l[1] = 42; // l is now @{x, 42, y, x+y@}
1743 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1747 You can append or prepend an expression to a list with the @code{append()}
1748 and @code{prepend()} methods:
1752 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1753 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1757 You can remove the first or last element of a list with @code{remove_first()}
1758 and @code{remove_last()}:
1762 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1763 l.remove_last(); // l is now @{x, 7, y, x+y@}
1767 You can remove all the elements of a list with @code{remove_all()}:
1771 l.remove_all(); // l is now empty
1775 You can bring the elements of a list into a canonical order with @code{sort()}:
1779 lst l1 = @{x, 2, y, x+y@};
1780 lst l2 = @{2, x+y, x, y@};
1783 // l1 and l2 are now equal
1787 Finally, you can remove all but the first element of consecutive groups of
1788 elements with @code{unique()}:
1792 lst l3 = @{x, 2, 2, 2, y, x+y, y+x@};
1793 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1798 @node Mathematical functions, Relations, Lists, Basic concepts
1799 @c node-name, next, previous, up
1800 @section Mathematical functions
1801 @cindex @code{function} (class)
1802 @cindex trigonometric function
1803 @cindex hyperbolic function
1805 There are quite a number of useful functions hard-wired into GiNaC. For
1806 instance, all trigonometric and hyperbolic functions are implemented
1807 (@xref{Built-in functions}, for a complete list).
1809 These functions (better called @emph{pseudofunctions}) are all objects
1810 of class @code{function}. They accept one or more expressions as
1811 arguments and return one expression. If the arguments are not
1812 numerical, the evaluation of the function may be halted, as it does in
1813 the next example, showing how a function returns itself twice and
1814 finally an expression that may be really useful:
1816 @cindex Gamma function
1817 @cindex @code{subs()}
1820 symbol x("x"), y("y");
1822 cout << tgamma(foo) << endl;
1823 // -> tgamma(x+(1/2)*y)
1824 ex bar = foo.subs(y==1);
1825 cout << tgamma(bar) << endl;
1827 ex foobar = bar.subs(x==7);
1828 cout << tgamma(foobar) << endl;
1829 // -> (135135/128)*Pi^(1/2)
1833 Besides evaluation most of these functions allow differentiation, series
1834 expansion and so on. Read the next chapter in order to learn more about
1837 It must be noted that these pseudofunctions are created by inline
1838 functions, where the argument list is templated. This means that
1839 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1840 @code{sin(ex(1))} and will therefore not result in a floating point
1841 number. Unless of course the function prototype is explicitly
1842 overridden -- which is the case for arguments of type @code{numeric}
1843 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1844 point number of class @code{numeric} you should call
1845 @code{sin(numeric(1))}. This is almost the same as calling
1846 @code{sin(1).evalf()} except that the latter will return a numeric
1847 wrapped inside an @code{ex}.
1850 @node Relations, Integrals, Mathematical functions, Basic concepts
1851 @c node-name, next, previous, up
1853 @cindex @code{relational} (class)
1855 Sometimes, a relation holding between two expressions must be stored
1856 somehow. The class @code{relational} is a convenient container for such
1857 purposes. A relation is by definition a container for two @code{ex} and
1858 a relation between them that signals equality, inequality and so on.
1859 They are created by simply using the C++ operators @code{==}, @code{!=},
1860 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1862 @xref{Mathematical functions}, for examples where various applications
1863 of the @code{.subs()} method show how objects of class relational are
1864 used as arguments. There they provide an intuitive syntax for
1865 substitutions. They are also used as arguments to the @code{ex::series}
1866 method, where the left hand side of the relation specifies the variable
1867 to expand in and the right hand side the expansion point. They can also
1868 be used for creating systems of equations that are to be solved for
1871 But the most common usage of objects of this class
1872 is rather inconspicuous in statements of the form @code{if
1873 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1874 conversion from @code{relational} to @code{bool} takes place. Note,
1875 however, that @code{==} here does not perform any simplifications, hence
1876 @code{expand()} must be called explicitly.
1879 relationals may be more efficient if preceded by a call to
1881 ex relational::canonical() const
1883 which returns an equivalent relation with the zero
1884 right-hand side. For example:
1887 relational rel = (p >= (p*p-1)/p);
1888 if (ex_to<relational>(rel.canonical().normal()))
1889 cout << "correct inequality" << endl;
1891 However, a user shall not expect that any inequality can be fully
1894 @node Integrals, Matrices, Relations, Basic concepts
1895 @c node-name, next, previous, up
1897 @cindex @code{integral} (class)
1899 An object of class @dfn{integral} can be used to hold a symbolic integral.
1900 If you want to symbolically represent the integral of @code{x*x} from 0 to
1901 1, you would write this as
1903 integral(x, 0, 1, x*x)
1905 The first argument is the integration variable. It should be noted that
1906 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1907 fact, it can only integrate polynomials. An expression containing integrals
1908 can be evaluated symbolically by calling the
1912 method on it. Numerical evaluation is available by calling the
1916 method on an expression containing the integral. This will only evaluate
1917 integrals into a number if @code{subs}ing the integration variable by a
1918 number in the fourth argument of an integral and then @code{evalf}ing the
1919 result always results in a number. Of course, also the boundaries of the
1920 integration domain must @code{evalf} into numbers. It should be noted that
1921 trying to @code{evalf} a function with discontinuities in the integration
1922 domain is not recommended. The accuracy of the numeric evaluation of
1923 integrals is determined by the static member variable
1925 ex integral::relative_integration_error
1927 of the class @code{integral}. The default value of this is 10^-8.
1928 The integration works by halving the interval of integration, until numeric
1929 stability of the answer indicates that the requested accuracy has been
1930 reached. The maximum depth of the halving can be set via the static member
1933 int integral::max_integration_level
1935 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1936 return the integral unevaluated. The function that performs the numerical
1937 evaluation, is also available as
1939 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1942 This function will throw an exception if the maximum depth is exceeded. The
1943 last parameter of the function is optional and defaults to the
1944 @code{relative_integration_error}. To make sure that we do not do too
1945 much work if an expression contains the same integral multiple times,
1946 a lookup table is used.
1948 If you know that an expression holds an integral, you can get the
1949 integration variable, the left boundary, right boundary and integrand by
1950 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1951 @code{.op(3)}. Differentiating integrals with respect to variables works
1952 as expected. Note that it makes no sense to differentiate an integral
1953 with respect to the integration variable.
1955 @node Matrices, Indexed objects, Integrals, Basic concepts
1956 @c node-name, next, previous, up
1958 @cindex @code{matrix} (class)
1960 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1961 matrix with @math{m} rows and @math{n} columns are accessed with two
1962 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1963 second one in the range 0@dots{}@math{n-1}.
1965 There are a couple of ways to construct matrices, with or without preset
1966 elements. The constructor
1969 matrix::matrix(unsigned r, unsigned c);
1972 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1975 The easiest way to create a matrix is using an initializer list of
1976 initializer lists, all of the same size:
1980 matrix m = @{@{1, -a@},
1985 You can also specify the elements as a (flat) list with
1988 matrix::matrix(unsigned r, unsigned c, const lst & l);
1993 @cindex @code{lst_to_matrix()}
1995 ex lst_to_matrix(const lst & l);
1998 constructs a matrix from a list of lists, each list representing a matrix row.
2000 There is also a set of functions for creating some special types of
2003 @cindex @code{diag_matrix()}
2004 @cindex @code{unit_matrix()}
2005 @cindex @code{symbolic_matrix()}
2007 ex diag_matrix(const lst & l);
2008 ex diag_matrix(initializer_list<ex> l);
2009 ex unit_matrix(unsigned x);
2010 ex unit_matrix(unsigned r, unsigned c);
2011 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
2012 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
2013 const string & tex_base_name);
2016 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
2017 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
2018 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
2019 matrix filled with newly generated symbols made of the specified base name
2020 and the position of each element in the matrix.
2022 Matrices often arise by omitting elements of another matrix. For
2023 instance, the submatrix @code{S} of a matrix @code{M} takes a
2024 rectangular block from @code{M}. The reduced matrix @code{R} is defined
2025 by removing one row and one column from a matrix @code{M}. (The
2026 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
2027 can be used for computing the inverse using Cramer's rule.)
2029 @cindex @code{sub_matrix()}
2030 @cindex @code{reduced_matrix()}
2032 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2033 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2036 The function @code{sub_matrix()} takes a row offset @code{r} and a
2037 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2038 columns. The function @code{reduced_matrix()} has two integer arguments
2039 that specify which row and column to remove:
2043 matrix m = @{@{11, 12, 13@},
2046 cout << reduced_matrix(m, 1, 1) << endl;
2047 // -> [[11,13],[31,33]]
2048 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2049 // -> [[22,23],[32,33]]
2053 Matrix elements can be accessed and set using the parenthesis (function call)
2057 const ex & matrix::operator()(unsigned r, unsigned c) const;
2058 ex & matrix::operator()(unsigned r, unsigned c);
2061 It is also possible to access the matrix elements in a linear fashion with
2062 the @code{op()} method. But C++-style subscripting with square brackets
2063 @samp{[]} is not available.
2065 Here are a couple of examples for constructing matrices:
2069 symbol a("a"), b("b");
2071 matrix M = @{@{a, 0@},
2082 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2085 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2088 cout << diag_matrix(lst@{a, b@}) << endl;
2091 cout << unit_matrix(3) << endl;
2092 // -> [[1,0,0],[0,1,0],[0,0,1]]
2094 cout << symbolic_matrix(2, 3, "x") << endl;
2095 // -> [[x00,x01,x02],[x10,x11,x12]]
2099 @cindex @code{is_zero_matrix()}
2100 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2101 all entries of the matrix are zeros. There is also method
2102 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2103 expression is zero or a zero matrix.
2105 @cindex @code{transpose()}
2106 There are three ways to do arithmetic with matrices. The first (and most
2107 direct one) is to use the methods provided by the @code{matrix} class:
2110 matrix matrix::add(const matrix & other) const;
2111 matrix matrix::sub(const matrix & other) const;
2112 matrix matrix::mul(const matrix & other) const;
2113 matrix matrix::mul_scalar(const ex & other) const;
2114 matrix matrix::pow(const ex & expn) const;
2115 matrix matrix::transpose() const;
2118 All of these methods return the result as a new matrix object. Here is an
2119 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2124 matrix A = @{@{ 1, 2@},
2126 matrix B = @{@{-1, 0@},
2128 matrix C = @{@{ 8, 4@},
2131 matrix result = A.mul(B).sub(C.mul_scalar(2));
2132 cout << result << endl;
2133 // -> [[-13,-6],[1,2]]
2138 @cindex @code{evalm()}
2139 The second (and probably the most natural) way is to construct an expression
2140 containing matrices with the usual arithmetic operators and @code{pow()}.
2141 For efficiency reasons, expressions with sums, products and powers of
2142 matrices are not automatically evaluated in GiNaC. You have to call the
2146 ex ex::evalm() const;
2149 to obtain the result:
2156 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2157 cout << e.evalm() << endl;
2158 // -> [[-13,-6],[1,2]]
2163 The non-commutativity of the product @code{A*B} in this example is
2164 automatically recognized by GiNaC. There is no need to use a special
2165 operator here. @xref{Non-commutative objects}, for more information about
2166 dealing with non-commutative expressions.
2168 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2169 to perform the arithmetic:
2174 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2175 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2177 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2178 cout << e.simplify_indexed() << endl;
2179 // -> [[-13,-6],[1,2]].i.j
2183 Using indices is most useful when working with rectangular matrices and
2184 one-dimensional vectors because you don't have to worry about having to
2185 transpose matrices before multiplying them. @xref{Indexed objects}, for
2186 more information about using matrices with indices, and about indices in
2189 The @code{matrix} class provides a couple of additional methods for
2190 computing determinants, traces, characteristic polynomials and ranks:
2192 @cindex @code{determinant()}
2193 @cindex @code{trace()}
2194 @cindex @code{charpoly()}
2195 @cindex @code{rank()}
2197 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2198 ex matrix::trace() const;
2199 ex matrix::charpoly(const ex & lambda) const;
2200 unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;
2203 The optional @samp{algo} argument of @code{determinant()} and @code{rank()}
2204 functions allows to select between different algorithms for calculating the
2205 determinant and rank respectively. The asymptotic speed (as parametrized
2206 by the matrix size) can greatly differ between those algorithms, depending
2207 on the nature of the matrix' entries. The possible values are defined in
2208 the @file{flags.h} header file. By default, GiNaC uses a heuristic to
2209 automatically select an algorithm that is likely (but not guaranteed)
2210 to give the result most quickly.
2212 @cindex @code{solve()}
2213 Linear systems can be solved with:
2216 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2217 unsigned algo=solve_algo::automatic) const;
2220 Assuming the matrix object this method is applied on is an @code{m}
2221 times @code{n} matrix, then @code{vars} must be a @code{n} times
2222 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2223 times @code{p} matrix. The returned matrix then has dimension @code{n}
2224 times @code{p} and in the case of an underdetermined system will still
2225 contain some of the indeterminates from @code{vars}. If the system is
2226 overdetermined, an exception is thrown.
2228 @cindex @code{inverse()} (matrix)
2229 To invert a matrix, use the method:
2232 matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
2235 The @samp{algo} argument is optional. If given, it must be one of
2236 @code{solve_algo} defined in @file{flags.h}.
2238 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2239 @c node-name, next, previous, up
2240 @section Indexed objects
2242 GiNaC allows you to handle expressions containing general indexed objects in
2243 arbitrary spaces. It is also able to canonicalize and simplify such
2244 expressions and perform symbolic dummy index summations. There are a number
2245 of predefined indexed objects provided, like delta and metric tensors.
2247 There are few restrictions placed on indexed objects and their indices and
2248 it is easy to construct nonsense expressions, but our intention is to
2249 provide a general framework that allows you to implement algorithms with
2250 indexed quantities, getting in the way as little as possible.
2252 @cindex @code{idx} (class)
2253 @cindex @code{indexed} (class)
2254 @subsection Indexed quantities and their indices
2256 Indexed expressions in GiNaC are constructed of two special types of objects,
2257 @dfn{index objects} and @dfn{indexed objects}.
2261 @cindex contravariant
2264 @item Index objects are of class @code{idx} or a subclass. Every index has
2265 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2266 the index lives in) which can both be arbitrary expressions but are usually
2267 a number or a simple symbol. In addition, indices of class @code{varidx} have
2268 a @dfn{variance} (they can be co- or contravariant), and indices of class
2269 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2271 @item Indexed objects are of class @code{indexed} or a subclass. They
2272 contain a @dfn{base expression} (which is the expression being indexed), and
2273 one or more indices.
2277 @strong{Please notice:} when printing expressions, covariant indices and indices
2278 without variance are denoted @samp{.i} while contravariant indices are
2279 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2280 value. In the following, we are going to use that notation in the text so
2281 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2282 not visible in the output.
2284 A simple example shall illustrate the concepts:
2288 #include <ginac/ginac.h>
2289 using namespace std;
2290 using namespace GiNaC;
2294 symbol i_sym("i"), j_sym("j");
2295 idx i(i_sym, 3), j(j_sym, 3);
2298 cout << indexed(A, i, j) << endl;
2300 cout << index_dimensions << indexed(A, i, j) << endl;
2302 cout << dflt; // reset cout to default output format (dimensions hidden)
2306 The @code{idx} constructor takes two arguments, the index value and the
2307 index dimension. First we define two index objects, @code{i} and @code{j},
2308 both with the numeric dimension 3. The value of the index @code{i} is the
2309 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2310 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2311 construct an expression containing one indexed object, @samp{A.i.j}. It has
2312 the symbol @code{A} as its base expression and the two indices @code{i} and
2315 The dimensions of indices are normally not visible in the output, but one
2316 can request them to be printed with the @code{index_dimensions} manipulator,
2319 Note the difference between the indices @code{i} and @code{j} which are of
2320 class @code{idx}, and the index values which are the symbols @code{i_sym}
2321 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2322 or numbers but must be index objects. For example, the following is not
2323 correct and will raise an exception:
2326 symbol i("i"), j("j");
2327 e = indexed(A, i, j); // ERROR: indices must be of type idx
2330 You can have multiple indexed objects in an expression, index values can
2331 be numeric, and index dimensions symbolic:
2335 symbol B("B"), dim("dim");
2336 cout << 4 * indexed(A, i)
2337 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2342 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2343 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2344 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2345 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2346 @code{simplify_indexed()} for that, see below).
2348 In fact, base expressions, index values and index dimensions can be
2349 arbitrary expressions:
2353 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2358 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2359 get an error message from this but you will probably not be able to do
2360 anything useful with it.
2362 @cindex @code{get_value()}
2363 @cindex @code{get_dim()}
2367 ex idx::get_value();
2371 return the value and dimension of an @code{idx} object. If you have an index
2372 in an expression, such as returned by calling @code{.op()} on an indexed
2373 object, you can get a reference to the @code{idx} object with the function
2374 @code{ex_to<idx>()} on the expression.
2376 There are also the methods
2379 bool idx::is_numeric();
2380 bool idx::is_symbolic();
2381 bool idx::is_dim_numeric();
2382 bool idx::is_dim_symbolic();
2385 for checking whether the value and dimension are numeric or symbolic
2386 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2387 about expressions}) returns information about the index value.
2389 @cindex @code{varidx} (class)
2390 If you need co- and contravariant indices, use the @code{varidx} class:
2394 symbol mu_sym("mu"), nu_sym("nu");
2395 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2396 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2398 cout << indexed(A, mu, nu) << endl;
2400 cout << indexed(A, mu_co, nu) << endl;
2402 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2407 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2408 co- or contravariant. The default is a contravariant (upper) index, but
2409 this can be overridden by supplying a third argument to the @code{varidx}
2410 constructor. The two methods
2413 bool varidx::is_covariant();
2414 bool varidx::is_contravariant();
2417 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2418 to get the object reference from an expression). There's also the very useful
2422 ex varidx::toggle_variance();
2425 which makes a new index with the same value and dimension but the opposite
2426 variance. By using it you only have to define the index once.
2428 @cindex @code{spinidx} (class)
2429 The @code{spinidx} class provides dotted and undotted variant indices, as
2430 used in the Weyl-van-der-Waerden spinor formalism:
2434 symbol K("K"), C_sym("C"), D_sym("D");
2435 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2436 // contravariant, undotted
2437 spinidx C_co(C_sym, 2, true); // covariant index
2438 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2439 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2441 cout << indexed(K, C, D) << endl;
2443 cout << indexed(K, C_co, D_dot) << endl;
2445 cout << indexed(K, D_co_dot, D) << endl;
2450 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2451 dotted or undotted. The default is undotted but this can be overridden by
2452 supplying a fourth argument to the @code{spinidx} constructor. The two
2456 bool spinidx::is_dotted();
2457 bool spinidx::is_undotted();
2460 allow you to check whether or not a @code{spinidx} object is dotted (use
2461 @code{ex_to<spinidx>()} to get the object reference from an expression).
2462 Finally, the two methods
2465 ex spinidx::toggle_dot();
2466 ex spinidx::toggle_variance_dot();
2469 create a new index with the same value and dimension but opposite dottedness
2470 and the same or opposite variance.
2472 @subsection Substituting indices
2474 @cindex @code{subs()}
2475 Sometimes you will want to substitute one symbolic index with another
2476 symbolic or numeric index, for example when calculating one specific element
2477 of a tensor expression. This is done with the @code{.subs()} method, as it
2478 is done for symbols (see @ref{Substituting expressions}).
2480 You have two possibilities here. You can either substitute the whole index
2481 by another index or expression:
2485 ex e = indexed(A, mu_co);
2486 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2487 // -> A.mu becomes A~nu
2488 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2489 // -> A.mu becomes A~0
2490 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2491 // -> A.mu becomes A.0
2495 The third example shows that trying to replace an index with something that
2496 is not an index will substitute the index value instead.
2498 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2503 ex e = indexed(A, mu_co);
2504 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2505 // -> A.mu becomes A.nu
2506 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2507 // -> A.mu becomes A.0
2511 As you see, with the second method only the value of the index will get
2512 substituted. Its other properties, including its dimension, remain unchanged.
2513 If you want to change the dimension of an index you have to substitute the
2514 whole index by another one with the new dimension.
2516 Finally, substituting the base expression of an indexed object works as
2521 ex e = indexed(A, mu_co);
2522 cout << e << " becomes " << e.subs(A == A+B) << endl;
2523 // -> A.mu becomes (B+A).mu
2527 @subsection Symmetries
2528 @cindex @code{symmetry} (class)
2529 @cindex @code{sy_none()}
2530 @cindex @code{sy_symm()}
2531 @cindex @code{sy_anti()}
2532 @cindex @code{sy_cycl()}
2534 Indexed objects can have certain symmetry properties with respect to their
2535 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2536 that is constructed with the helper functions
2539 symmetry sy_none(...);
2540 symmetry sy_symm(...);
2541 symmetry sy_anti(...);
2542 symmetry sy_cycl(...);
2545 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2546 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2547 represents a cyclic symmetry. Each of these functions accepts up to four
2548 arguments which can be either symmetry objects themselves or unsigned integer
2549 numbers that represent an index position (counting from 0). A symmetry
2550 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2551 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2554 Here are some examples of symmetry definitions:
2559 e = indexed(A, i, j);
2560 e = indexed(A, sy_none(), i, j); // equivalent
2561 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2563 // Symmetric in all three indices:
2564 e = indexed(A, sy_symm(), i, j, k);
2565 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2566 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2567 // different canonical order
2569 // Symmetric in the first two indices only:
2570 e = indexed(A, sy_symm(0, 1), i, j, k);
2571 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2573 // Antisymmetric in the first and last index only (index ranges need not
2575 e = indexed(A, sy_anti(0, 2), i, j, k);
2576 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2578 // An example of a mixed symmetry: antisymmetric in the first two and
2579 // last two indices, symmetric when swapping the first and last index
2580 // pairs (like the Riemann curvature tensor):
2581 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2583 // Cyclic symmetry in all three indices:
2584 e = indexed(A, sy_cycl(), i, j, k);
2585 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2587 // The following examples are invalid constructions that will throw
2588 // an exception at run time.
2590 // An index may not appear multiple times:
2591 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2592 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2594 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2595 // same number of indices:
2596 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2598 // And of course, you cannot specify indices which are not there:
2599 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2603 If you need to specify more than four indices, you have to use the
2604 @code{.add()} method of the @code{symmetry} class. For example, to specify
2605 full symmetry in the first six indices you would write
2606 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2608 If an indexed object has a symmetry, GiNaC will automatically bring the
2609 indices into a canonical order which allows for some immediate simplifications:
2613 cout << indexed(A, sy_symm(), i, j)
2614 + indexed(A, sy_symm(), j, i) << endl;
2616 cout << indexed(B, sy_anti(), i, j)
2617 + indexed(B, sy_anti(), j, i) << endl;
2619 cout << indexed(B, sy_anti(), i, j, k)
2620 - indexed(B, sy_anti(), j, k, i) << endl;
2625 @cindex @code{get_free_indices()}
2627 @subsection Dummy indices
2629 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2630 that a summation over the index range is implied. Symbolic indices which are
2631 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2632 dummy nor free indices.
2634 To be recognized as a dummy index pair, the two indices must be of the same
2635 class and their value must be the same single symbol (an index like
2636 @samp{2*n+1} is never a dummy index). If the indices are of class
2637 @code{varidx} they must also be of opposite variance; if they are of class
2638 @code{spinidx} they must be both dotted or both undotted.
2640 The method @code{.get_free_indices()} returns a vector containing the free
2641 indices of an expression. It also checks that the free indices of the terms
2642 of a sum are consistent:
2646 symbol A("A"), B("B"), C("C");
2648 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2649 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2651 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2652 cout << exprseq(e.get_free_indices()) << endl;
2654 // 'j' and 'l' are dummy indices
2656 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2657 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2659 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2660 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2661 cout << exprseq(e.get_free_indices()) << endl;
2663 // 'nu' is a dummy index, but 'sigma' is not
2665 e = indexed(A, mu, mu);
2666 cout << exprseq(e.get_free_indices()) << endl;
2668 // 'mu' is not a dummy index because it appears twice with the same
2671 e = indexed(A, mu, nu) + 42;
2672 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2673 // this will throw an exception:
2674 // "add::get_free_indices: inconsistent indices in sum"
2678 @cindex @code{expand_dummy_sum()}
2679 A dummy index summation like
2686 can be expanded for indices with numeric
2687 dimensions (e.g. 3) into the explicit sum like
2689 $a_1b^1+a_2b^2+a_3b^3 $.
2692 a.1 b~1 + a.2 b~2 + a.3 b~3.
2694 This is performed by the function
2697 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2700 which takes an expression @code{e} and returns the expanded sum for all
2701 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2702 is set to @code{true} then all substitutions are made by @code{idx} class
2703 indices, i.e. without variance. In this case the above sum
2712 $a_1b_1+a_2b_2+a_3b_3 $.
2715 a.1 b.1 + a.2 b.2 + a.3 b.3.
2719 @cindex @code{simplify_indexed()}
2720 @subsection Simplifying indexed expressions
2722 In addition to the few automatic simplifications that GiNaC performs on
2723 indexed expressions (such as re-ordering the indices of symmetric tensors
2724 and calculating traces and convolutions of matrices and predefined tensors)
2728 ex ex::simplify_indexed();
2729 ex ex::simplify_indexed(const scalar_products & sp);
2732 that performs some more expensive operations:
2735 @item it checks the consistency of free indices in sums in the same way
2736 @code{get_free_indices()} does
2737 @item it tries to give dummy indices that appear in different terms of a sum
2738 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2739 @item it (symbolically) calculates all possible dummy index summations/contractions
2740 with the predefined tensors (this will be explained in more detail in the
2742 @item it detects contractions that vanish for symmetry reasons, for example
2743 the contraction of a symmetric and a totally antisymmetric tensor
2744 @item as a special case of dummy index summation, it can replace scalar products
2745 of two tensors with a user-defined value
2748 The last point is done with the help of the @code{scalar_products} class
2749 which is used to store scalar products with known values (this is not an
2750 arithmetic class, you just pass it to @code{simplify_indexed()}):
2754 symbol A("A"), B("B"), C("C"), i_sym("i");
2758 sp.add(A, B, 0); // A and B are orthogonal
2759 sp.add(A, C, 0); // A and C are orthogonal
2760 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2762 e = indexed(A + B, i) * indexed(A + C, i);
2764 // -> (B+A).i*(A+C).i
2766 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2772 The @code{scalar_products} object @code{sp} acts as a storage for the
2773 scalar products added to it with the @code{.add()} method. This method
2774 takes three arguments: the two expressions of which the scalar product is
2775 taken, and the expression to replace it with.
2777 @cindex @code{expand()}
2778 The example above also illustrates a feature of the @code{expand()} method:
2779 if passed the @code{expand_indexed} option it will distribute indices
2780 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2782 @cindex @code{tensor} (class)
2783 @subsection Predefined tensors
2785 Some frequently used special tensors such as the delta, epsilon and metric
2786 tensors are predefined in GiNaC. They have special properties when
2787 contracted with other tensor expressions and some of them have constant
2788 matrix representations (they will evaluate to a number when numeric
2789 indices are specified).
2791 @cindex @code{delta_tensor()}
2792 @subsubsection Delta tensor
2794 The delta tensor takes two indices, is symmetric and has the matrix
2795 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2796 @code{delta_tensor()}:
2800 symbol A("A"), B("B");
2802 idx i(symbol("i"), 3), j(symbol("j"), 3),
2803 k(symbol("k"), 3), l(symbol("l"), 3);
2805 ex e = indexed(A, i, j) * indexed(B, k, l)
2806 * delta_tensor(i, k) * delta_tensor(j, l);
2807 cout << e.simplify_indexed() << endl;
2810 cout << delta_tensor(i, i) << endl;
2815 @cindex @code{metric_tensor()}
2816 @subsubsection General metric tensor
2818 The function @code{metric_tensor()} creates a general symmetric metric
2819 tensor with two indices that can be used to raise/lower tensor indices. The
2820 metric tensor is denoted as @samp{g} in the output and if its indices are of
2821 mixed variance it is automatically replaced by a delta tensor:
2827 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2829 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2830 cout << e.simplify_indexed() << endl;
2833 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2834 cout << e.simplify_indexed() << endl;
2837 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2838 * metric_tensor(nu, rho);
2839 cout << e.simplify_indexed() << endl;
2842 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2843 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2844 + indexed(A, mu.toggle_variance(), rho));
2845 cout << e.simplify_indexed() << endl;
2850 @cindex @code{lorentz_g()}
2851 @subsubsection Minkowski metric tensor
2853 The Minkowski metric tensor is a special metric tensor with a constant
2854 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2855 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2856 It is created with the function @code{lorentz_g()} (although it is output as
2861 varidx mu(symbol("mu"), 4);
2863 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2864 * lorentz_g(mu, varidx(0, 4)); // negative signature
2865 cout << e.simplify_indexed() << endl;
2868 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2869 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2870 cout << e.simplify_indexed() << endl;
2875 @cindex @code{spinor_metric()}
2876 @subsubsection Spinor metric tensor
2878 The function @code{spinor_metric()} creates an antisymmetric tensor with
2879 two indices that is used to raise/lower indices of 2-component spinors.
2880 It is output as @samp{eps}:
2886 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2887 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2889 e = spinor_metric(A, B) * indexed(psi, B_co);
2890 cout << e.simplify_indexed() << endl;
2893 e = spinor_metric(A, B) * indexed(psi, A_co);
2894 cout << e.simplify_indexed() << endl;
2897 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2898 cout << e.simplify_indexed() << endl;
2901 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2902 cout << e.simplify_indexed() << endl;
2905 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2906 cout << e.simplify_indexed() << endl;
2909 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2910 cout << e.simplify_indexed() << endl;
2915 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2917 @cindex @code{epsilon_tensor()}
2918 @cindex @code{lorentz_eps()}
2919 @subsubsection Epsilon tensor
2921 The epsilon tensor is totally antisymmetric, its number of indices is equal
2922 to the dimension of the index space (the indices must all be of the same
2923 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2924 defined to be 1. Its behavior with indices that have a variance also
2925 depends on the signature of the metric. Epsilon tensors are output as
2928 There are three functions defined to create epsilon tensors in 2, 3 and 4
2932 ex epsilon_tensor(const ex & i1, const ex & i2);
2933 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2934 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2935 bool pos_sig = false);
2938 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2939 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2940 Minkowski space (the last @code{bool} argument specifies whether the metric
2941 has negative or positive signature, as in the case of the Minkowski metric
2946 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2947 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2948 e = lorentz_eps(mu, nu, rho, sig) *
2949 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2950 cout << simplify_indexed(e) << endl;
2951 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2953 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2954 symbol A("A"), B("B");
2955 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2956 cout << simplify_indexed(e) << endl;
2957 // -> -B.k*A.j*eps.i.k.j
2958 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2959 cout << simplify_indexed(e) << endl;
2964 @subsection Linear algebra
2966 The @code{matrix} class can be used with indices to do some simple linear
2967 algebra (linear combinations and products of vectors and matrices, traces
2968 and scalar products):
2972 idx i(symbol("i"), 2), j(symbol("j"), 2);
2973 symbol x("x"), y("y");
2975 // A is a 2x2 matrix, X is a 2x1 vector
2976 matrix A = @{@{1, 2@},
2978 matrix X = @{@{x, y@}@};
2980 cout << indexed(A, i, i) << endl;
2983 ex e = indexed(A, i, j) * indexed(X, j);
2984 cout << e.simplify_indexed() << endl;
2985 // -> [[2*y+x],[4*y+3*x]].i
2987 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2988 cout << e.simplify_indexed() << endl;
2989 // -> [[3*y+3*x,6*y+2*x]].j
2993 You can of course obtain the same results with the @code{matrix::add()},
2994 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2995 but with indices you don't have to worry about transposing matrices.
2997 Matrix indices always start at 0 and their dimension must match the number
2998 of rows/columns of the matrix. Matrices with one row or one column are
2999 vectors and can have one or two indices (it doesn't matter whether it's a
3000 row or a column vector). Other matrices must have two indices.
3002 You should be careful when using indices with variance on matrices. GiNaC
3003 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
3004 @samp{F.mu.nu} are different matrices. In this case you should use only
3005 one form for @samp{F} and explicitly multiply it with a matrix representation
3006 of the metric tensor.
3009 @node Non-commutative objects, Methods and functions, Indexed objects, Basic concepts
3010 @c node-name, next, previous, up
3011 @section Non-commutative objects
3013 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
3014 non-commutative objects are built-in which are mostly of use in high energy
3018 @item Clifford (Dirac) algebra (class @code{clifford})
3019 @item su(3) Lie algebra (class @code{color})
3020 @item Matrices (unindexed) (class @code{matrix})
3023 The @code{clifford} and @code{color} classes are subclasses of
3024 @code{indexed} because the elements of these algebras usually carry
3025 indices. The @code{matrix} class is described in more detail in
3028 Unlike most computer algebra systems, GiNaC does not primarily provide an
3029 operator (often denoted @samp{&*}) for representing inert products of
3030 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
3031 classes of objects involved, and non-commutative products are formed with
3032 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3033 figuring out by itself which objects commutate and will group the factors
3034 by their class. Consider this example:
3038 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3039 idx a(symbol("a"), 8), b(symbol("b"), 8);
3040 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3042 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3046 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3047 groups the non-commutative factors (the gammas and the su(3) generators)
3048 together while preserving the order of factors within each class (because
3049 Clifford objects commutate with color objects). The resulting expression is a
3050 @emph{commutative} product with two factors that are themselves non-commutative
3051 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3052 parentheses are placed around the non-commutative products in the output.
3054 @cindex @code{ncmul} (class)
3055 Non-commutative products are internally represented by objects of the class
3056 @code{ncmul}, as opposed to commutative products which are handled by the
3057 @code{mul} class. You will normally not have to worry about this distinction,
3060 The advantage of this approach is that you never have to worry about using
3061 (or forgetting to use) a special operator when constructing non-commutative
3062 expressions. Also, non-commutative products in GiNaC are more intelligent
3063 than in other computer algebra systems; they can, for example, automatically
3064 canonicalize themselves according to rules specified in the implementation
3065 of the non-commutative classes. The drawback is that to work with other than
3066 the built-in algebras you have to implement new classes yourself. Both
3067 symbols and user-defined functions can be specified as being non-commutative.
3068 For symbols, this is done by subclassing class symbol; for functions,
3069 by explicitly setting the return type (@pxref{Symbolic functions}).
3071 @cindex @code{return_type()}
3072 @cindex @code{return_type_tinfo()}
3073 Information about the commutativity of an object or expression can be
3074 obtained with the two member functions
3077 unsigned ex::return_type() const;
3078 return_type_t ex::return_type_tinfo() const;
3081 The @code{return_type()} function returns one of three values (defined in
3082 the header file @file{flags.h}), corresponding to three categories of
3083 expressions in GiNaC:
3086 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3087 classes are of this kind.
3088 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3089 certain class of non-commutative objects which can be determined with the
3090 @code{return_type_tinfo()} method. Expressions of this category commutate
3091 with everything except @code{noncommutative} expressions of the same
3093 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3094 of non-commutative objects of different classes. Expressions of this
3095 category don't commutate with any other @code{noncommutative} or
3096 @code{noncommutative_composite} expressions.
3099 The @code{return_type_tinfo()} method returns an object of type
3100 @code{return_type_t} that contains information about the type of the expression
3101 and, if given, its representation label (see section on dirac gamma matrices for
3102 more details). The objects of type @code{return_type_t} can be tested for
3103 equality to test whether two expressions belong to the same category and
3104 therefore may not commute.
3106 Here are a couple of examples:
3109 @multitable @columnfractions .6 .4
3110 @item @strong{Expression} @tab @strong{@code{return_type()}}
3111 @item @code{42} @tab @code{commutative}
3112 @item @code{2*x-y} @tab @code{commutative}
3113 @item @code{dirac_ONE()} @tab @code{noncommutative}
3114 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3115 @item @code{2*color_T(a)} @tab @code{noncommutative}
3116 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3120 A last note: With the exception of matrices, positive integer powers of
3121 non-commutative objects are automatically expanded in GiNaC. For example,
3122 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3123 non-commutative expressions).
3126 @cindex @code{clifford} (class)
3127 @subsection Clifford algebra
3130 Clifford algebras are supported in two flavours: Dirac gamma
3131 matrices (more physical) and generic Clifford algebras (more
3134 @cindex @code{dirac_gamma()}
3135 @subsubsection Dirac gamma matrices
3136 Dirac gamma matrices (note that GiNaC doesn't treat them
3137 as matrices) are designated as @samp{gamma~mu} and satisfy
3138 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3139 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3140 constructed by the function
3143 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3146 which takes two arguments: the index and a @dfn{representation label} in the
3147 range 0 to 255 which is used to distinguish elements of different Clifford
3148 algebras (this is also called a @dfn{spin line index}). Gammas with different
3149 labels commutate with each other. The dimension of the index can be 4 or (in
3150 the framework of dimensional regularization) any symbolic value. Spinor
3151 indices on Dirac gammas are not supported in GiNaC.
3153 @cindex @code{dirac_ONE()}
3154 The unity element of a Clifford algebra is constructed by
3157 ex dirac_ONE(unsigned char rl = 0);
3160 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3161 multiples of the unity element, even though it's customary to omit it.
3162 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3163 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3164 GiNaC will complain and/or produce incorrect results.
3166 @cindex @code{dirac_gamma5()}
3167 There is a special element @samp{gamma5} that commutates with all other
3168 gammas, has a unit square, and in 4 dimensions equals
3169 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3172 ex dirac_gamma5(unsigned char rl = 0);
3175 @cindex @code{dirac_gammaL()}
3176 @cindex @code{dirac_gammaR()}
3177 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3178 objects, constructed by
3181 ex dirac_gammaL(unsigned char rl = 0);
3182 ex dirac_gammaR(unsigned char rl = 0);
3185 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3186 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3188 @cindex @code{dirac_slash()}
3189 Finally, the function
3192 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3195 creates a term that represents a contraction of @samp{e} with the Dirac
3196 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3197 with a unique index whose dimension is given by the @code{dim} argument).
3198 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3200 In products of dirac gammas, superfluous unity elements are automatically
3201 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3202 and @samp{gammaR} are moved to the front.
3204 The @code{simplify_indexed()} function performs contractions in gamma strings,
3210 symbol a("a"), b("b"), D("D");
3211 varidx mu(symbol("mu"), D);
3212 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3213 * dirac_gamma(mu.toggle_variance());
3215 // -> gamma~mu*a\*gamma.mu
3216 e = e.simplify_indexed();
3219 cout << e.subs(D == 4) << endl;
3225 @cindex @code{dirac_trace()}
3226 To calculate the trace of an expression containing strings of Dirac gammas
3227 you use one of the functions
3230 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3231 const ex & trONE = 4);
3232 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3233 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3236 These functions take the trace over all gammas in the specified set @code{rls}
3237 or list @code{rll} of representation labels, or the single label @code{rl};
3238 gammas with other labels are left standing. The last argument to
3239 @code{dirac_trace()} is the value to be returned for the trace of the unity
3240 element, which defaults to 4.
3242 The @code{dirac_trace()} function is a linear functional that is equal to the
3243 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3244 functional is not cyclic in
3251 dimensions when acting on
3252 expressions containing @samp{gamma5}, so it's not a proper trace. This
3253 @samp{gamma5} scheme is described in greater detail in the article
3254 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3256 The value of the trace itself is also usually different in 4 and in
3268 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3269 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3270 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3271 cout << dirac_trace(e).simplify_indexed() << endl;
3278 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3279 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3280 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3281 cout << dirac_trace(e).simplify_indexed() << endl;
3282 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3286 Here is an example for using @code{dirac_trace()} to compute a value that
3287 appears in the calculation of the one-loop vacuum polarization amplitude in
3292 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3293 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3296 sp.add(l, l, pow(l, 2));
3297 sp.add(l, q, ldotq);
3299 ex e = dirac_gamma(mu) *
3300 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3301 dirac_gamma(mu.toggle_variance()) *
3302 (dirac_slash(l, D) + m * dirac_ONE());
3303 e = dirac_trace(e).simplify_indexed(sp);
3304 e = e.collect(lst@{l, ldotq, m@});
3306 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3310 The @code{canonicalize_clifford()} function reorders all gamma products that
3311 appear in an expression to a canonical (but not necessarily simple) form.
3312 You can use this to compare two expressions or for further simplifications:
3316 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3317 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3319 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3321 e = canonicalize_clifford(e);
3323 // -> 2*ONE*eta~mu~nu
3327 @cindex @code{clifford_unit()}
3328 @subsubsection A generic Clifford algebra
3330 A generic Clifford algebra, i.e. a
3337 dimensional algebra with generators
3344 satisfying the identities
3346 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3349 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3351 for some bilinear form (@code{metric})
3352 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3353 and contain symbolic entries. Such generators are created by the
3357 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3360 where @code{mu} should be a @code{idx} (or descendant) class object
3361 indexing the generators.
3362 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3363 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3364 object. In fact, any expression either with two free indices or without
3365 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3366 object with two newly created indices with @code{metr} as its
3367 @code{op(0)} will be used.
3368 Optional parameter @code{rl} allows to distinguish different
3369 Clifford algebras, which will commute with each other.
3371 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3372 something very close to @code{dirac_gamma(mu)}, although
3373 @code{dirac_gamma} have more efficient simplification mechanism.
3374 @cindex @code{get_metric()}
3375 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3376 not aware about the symmetry of its metric, see the start of the previous
3377 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3378 specifies as follows:
3381 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3384 The method @code{clifford::get_metric()} returns a metric defining this
3387 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3388 the Clifford algebra units with a call like that
3391 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3394 since this may yield some further automatic simplifications. Again, for a
3395 metric defined through a @code{matrix} such a symmetry is detected
3398 Individual generators of a Clifford algebra can be accessed in several
3404 idx i(symbol("i"), 4);
3406 ex M = diag_matrix(lst@{1, -1, 0, s@});
3407 ex e = clifford_unit(i, M);
3408 ex e0 = e.subs(i == 0);
3409 ex e1 = e.subs(i == 1);
3410 ex e2 = e.subs(i == 2);
3411 ex e3 = e.subs(i == 3);
3416 will produce four anti-commuting generators of a Clifford algebra with properties
3418 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3421 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3422 @code{pow(e3, 2) = s}.
3425 @cindex @code{lst_to_clifford()}
3426 A similar effect can be achieved from the function
3429 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3430 unsigned char rl = 0);
3431 ex lst_to_clifford(const ex & v, const ex & e);
3434 which converts a list or vector
3436 $v = (v^0, v^1, ..., v^n)$
3439 @samp{v = (v~0, v~1, ..., v~n)}
3444 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3447 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3450 directly supplied in the second form of the procedure. In the first form
3451 the Clifford unit @samp{e.k} is generated by the call of
3452 @code{clifford_unit(mu, metr, rl)}.
3453 @cindex pseudo-vector
3454 If the number of components supplied
3455 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3456 1 then function @code{lst_to_clifford()} uses the following
3457 pseudo-vector representation:
3459 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3462 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3465 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3470 idx i(symbol("i"), 4);
3472 ex M = diag_matrix(@{1, -1, 0, s@});
3473 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3474 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3475 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3476 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3481 @cindex @code{clifford_to_lst()}
3482 There is the inverse function
3485 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3488 which takes an expression @code{e} and tries to find a list
3490 $v = (v^0, v^1, ..., v^n)$
3493 @samp{v = (v~0, v~1, ..., v~n)}
3495 such that the expression is either vector
3497 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3500 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3504 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3507 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3509 with respect to the given Clifford units @code{c}. Here none of the
3510 @samp{v~k} should contain Clifford units @code{c} (of course, this
3511 may be impossible). This function can use an @code{algebraic} method
3512 (default) or a symbolic one. With the @code{algebraic} method the
3513 @samp{v~k} are calculated as
3515 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3518 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3520 is zero or is not @code{numeric} for some @samp{k}
3521 then the method will be automatically changed to symbolic. The same effect
3522 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3524 @cindex @code{clifford_prime()}
3525 @cindex @code{clifford_star()}
3526 @cindex @code{clifford_bar()}
3527 There are several functions for (anti-)automorphisms of Clifford algebras:
3530 ex clifford_prime(const ex & e)
3531 inline ex clifford_star(const ex & e)
3532 inline ex clifford_bar(const ex & e)
3535 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3536 changes signs of all Clifford units in the expression. The reversion
3537 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3538 units in any product. Finally the main anti-automorphism
3539 of a Clifford algebra @code{clifford_bar()} is the composition of the
3540 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3541 in a product. These functions correspond to the notations
3556 used in Clifford algebra textbooks.
3558 @cindex @code{clifford_norm()}
3562 ex clifford_norm(const ex & e);
3565 @cindex @code{clifford_inverse()}
3566 calculates the norm of a Clifford number from the expression
3568 $||e||^2 = e\overline{e}$.
3571 @code{||e||^2 = e \bar@{e@}}
3573 The inverse of a Clifford expression is returned by the function
3576 ex clifford_inverse(const ex & e);
3579 which calculates it as
3581 $e^{-1} = \overline{e}/||e||^2$.
3584 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3593 then an exception is raised.
3595 @cindex @code{remove_dirac_ONE()}
3596 If a Clifford number happens to be a factor of
3597 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3598 expression by the function
3601 ex remove_dirac_ONE(const ex & e);
3604 @cindex @code{canonicalize_clifford()}
3605 The function @code{canonicalize_clifford()} works for a
3606 generic Clifford algebra in a similar way as for Dirac gammas.
3608 The next provided function is
3610 @cindex @code{clifford_moebius_map()}
3612 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3613 const ex & d, const ex & v, const ex & G,
3614 unsigned char rl = 0);
3615 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3616 unsigned char rl = 0);
3619 It takes a list or vector @code{v} and makes the Moebius (conformal or
3620 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3621 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3622 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3623 indexed object, tensormetric, matrix or a Clifford unit, in the later
3624 case the optional parameter @code{rl} is ignored even if supplied.
3625 Depending from the type of @code{v} the returned value of this function
3626 is either a vector or a list holding vector's components.
3628 @cindex @code{clifford_max_label()}
3629 Finally the function
3632 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3635 can detect a presence of Clifford objects in the expression @code{e}: if
3636 such objects are found it returns the maximal
3637 @code{representation_label} of them, otherwise @code{-1}. The optional
3638 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3639 be ignored during the search.
3641 LaTeX output for Clifford units looks like
3642 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3643 @code{representation_label} and @code{\nu} is the index of the
3644 corresponding unit. This provides a flexible typesetting with a suitable
3645 definition of the @code{\clifford} command. For example, the definition
3647 \newcommand@{\clifford@}[1][]@{@}
3649 typesets all Clifford units identically, while the alternative definition
3651 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3653 prints units with @code{representation_label=0} as
3660 with @code{representation_label=1} as
3667 and with @code{representation_label=2} as
3675 @cindex @code{color} (class)
3676 @subsection Color algebra
3678 @cindex @code{color_T()}
3679 For computations in quantum chromodynamics, GiNaC implements the base elements
3680 and structure constants of the su(3) Lie algebra (color algebra). The base
3681 elements @math{T_a} are constructed by the function
3684 ex color_T(const ex & a, unsigned char rl = 0);
3687 which takes two arguments: the index and a @dfn{representation label} in the
3688 range 0 to 255 which is used to distinguish elements of different color
3689 algebras. Objects with different labels commutate with each other. The
3690 dimension of the index must be exactly 8 and it should be of class @code{idx},
3693 @cindex @code{color_ONE()}
3694 The unity element of a color algebra is constructed by
3697 ex color_ONE(unsigned char rl = 0);
3700 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3701 multiples of the unity element, even though it's customary to omit it.
3702 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3703 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3704 GiNaC may produce incorrect results.
3706 @cindex @code{color_d()}
3707 @cindex @code{color_f()}
3711 ex color_d(const ex & a, const ex & b, const ex & c);
3712 ex color_f(const ex & a, const ex & b, const ex & c);
3715 create the symmetric and antisymmetric structure constants @math{d_abc} and
3716 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3717 and @math{[T_a, T_b] = i f_abc T_c}.
3719 These functions evaluate to their numerical values,
3720 if you supply numeric indices to them. The index values should be in
3721 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3722 goes along better with the notations used in physical literature.
3724 @cindex @code{color_h()}
3725 There's an additional function
3728 ex color_h(const ex & a, const ex & b, const ex & c);
3731 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3733 The function @code{simplify_indexed()} performs some simplifications on
3734 expressions containing color objects:
3739 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3740 k(symbol("k"), 8), l(symbol("l"), 8);
3742 e = color_d(a, b, l) * color_f(a, b, k);
3743 cout << e.simplify_indexed() << endl;
3746 e = color_d(a, b, l) * color_d(a, b, k);
3747 cout << e.simplify_indexed() << endl;
3750 e = color_f(l, a, b) * color_f(a, b, k);
3751 cout << e.simplify_indexed() << endl;
3754 e = color_h(a, b, c) * color_h(a, b, c);
3755 cout << e.simplify_indexed() << endl;
3758 e = color_h(a, b, c) * color_T(b) * color_T(c);
3759 cout << e.simplify_indexed() << endl;
3762 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3763 cout << e.simplify_indexed() << endl;
3766 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3767 cout << e.simplify_indexed() << endl;
3768 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3772 @cindex @code{color_trace()}
3773 To calculate the trace of an expression containing color objects you use one
3777 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3778 ex color_trace(const ex & e, const lst & rll);
3779 ex color_trace(const ex & e, unsigned char rl = 0);
3782 These functions take the trace over all color @samp{T} objects in the
3783 specified set @code{rls} or list @code{rll} of representation labels, or the
3784 single label @code{rl}; @samp{T}s with other labels are left standing. For
3789 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3791 // -> -I*f.a.c.b+d.a.c.b
3796 @node Methods and functions, Information about expressions, Non-commutative objects, Top
3797 @c node-name, next, previous, up
3798 @chapter Methods and functions
3801 In this chapter the most important algorithms provided by GiNaC will be
3802 described. Some of them are implemented as functions on expressions,
3803 others are implemented as methods provided by expression objects. If
3804 they are methods, there exists a wrapper function around it, so you can
3805 alternatively call it in a functional way as shown in the simple
3810 cout << "As method: " << sin(1).evalf() << endl;
3811 cout << "As function: " << evalf(sin(1)) << endl;
3815 @cindex @code{subs()}
3816 The general rule is that wherever methods accept one or more parameters
3817 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3818 wrapper accepts is the same but preceded by the object to act on
3819 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3820 most natural one in an OO model but it may lead to confusion for MapleV
3821 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3822 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3823 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3824 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3825 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3826 here. Also, users of MuPAD will in most cases feel more comfortable
3827 with GiNaC's convention. All function wrappers are implemented
3828 as simple inline functions which just call the corresponding method and
3829 are only provided for users uncomfortable with OO who are dead set to
3830 avoid method invocations. Generally, nested function wrappers are much
3831 harder to read than a sequence of methods and should therefore be
3832 avoided if possible. On the other hand, not everything in GiNaC is a
3833 method on class @code{ex} and sometimes calling a function cannot be
3837 * Information about expressions::
3838 * Numerical evaluation::
3839 * Substituting expressions::
3840 * Pattern matching and advanced substitutions::
3841 * Applying a function on subexpressions::
3842 * Visitors and tree traversal::
3843 * Polynomial arithmetic:: Working with polynomials.
3844 * Rational expressions:: Working with rational functions.
3845 * Symbolic differentiation::
3846 * Series expansion:: Taylor and Laurent expansion.
3848 * Built-in functions:: List of predefined mathematical functions.
3849 * Multiple polylogarithms::
3850 * Iterated integrals::
3851 * Complex expressions::
3852 * Solving linear systems of equations::
3853 * Input/output:: Input and output of expressions.
3857 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3858 @c node-name, next, previous, up
3859 @section Getting information about expressions
3861 @subsection Checking expression types
3862 @cindex @code{is_a<@dots{}>()}
3863 @cindex @code{is_exactly_a<@dots{}>()}
3864 @cindex @code{ex_to<@dots{}>()}
3865 @cindex Converting @code{ex} to other classes
3866 @cindex @code{info()}
3867 @cindex @code{return_type()}
3868 @cindex @code{return_type_tinfo()}
3870 Sometimes it's useful to check whether a given expression is a plain number,
3871 a sum, a polynomial with integer coefficients, or of some other specific type.
3872 GiNaC provides a couple of functions for this:
3875 bool is_a<T>(const ex & e);
3876 bool is_exactly_a<T>(const ex & e);
3877 bool ex::info(unsigned flag);
3878 unsigned ex::return_type() const;
3879 return_type_t ex::return_type_tinfo() const;
3882 When the test made by @code{is_a<T>()} returns true, it is safe to call
3883 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3884 class names (@xref{The class hierarchy}, for a list of all classes). For
3885 example, assuming @code{e} is an @code{ex}:
3890 if (is_a<numeric>(e))
3891 numeric n = ex_to<numeric>(e);
3896 @code{is_a<T>(e)} allows you to check whether the top-level object of
3897 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3898 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3899 e.g., for checking whether an expression is a number, a sum, or a product:
3906 is_a<numeric>(e1); // true
3907 is_a<numeric>(e2); // false
3908 is_a<add>(e1); // false
3909 is_a<add>(e2); // true
3910 is_a<mul>(e1); // false
3911 is_a<mul>(e2); // false
3915 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3916 top-level object of an expression @samp{e} is an instance of the GiNaC
3917 class @samp{T}, not including parent classes.
3919 The @code{info()} method is used for checking certain attributes of
3920 expressions. The possible values for the @code{flag} argument are defined
3921 in @file{ginac/flags.h}, the most important being explained in the following
3925 @multitable @columnfractions .30 .70
3926 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3927 @item @code{numeric}
3928 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3930 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3931 @item @code{rational}
3932 @tab @dots{}an exact rational number (integers are rational, too)
3933 @item @code{integer}
3934 @tab @dots{}a (non-complex) integer
3935 @item @code{crational}
3936 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3937 @item @code{cinteger}
3938 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3939 @item @code{positive}
3940 @tab @dots{}not complex and greater than 0
3941 @item @code{negative}
3942 @tab @dots{}not complex and less than 0
3943 @item @code{nonnegative}
3944 @tab @dots{}not complex and greater than or equal to 0
3946 @tab @dots{}an integer greater than 0
3948 @tab @dots{}an integer less than 0
3949 @item @code{nonnegint}
3950 @tab @dots{}an integer greater than or equal to 0
3952 @tab @dots{}an even integer
3954 @tab @dots{}an odd integer
3956 @tab @dots{}a prime integer (probabilistic primality test)
3957 @item @code{relation}
3958 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3959 @item @code{relation_equal}
3960 @tab @dots{}a @code{==} relation
3961 @item @code{relation_not_equal}
3962 @tab @dots{}a @code{!=} relation
3963 @item @code{relation_less}
3964 @tab @dots{}a @code{<} relation
3965 @item @code{relation_less_or_equal}
3966 @tab @dots{}a @code{<=} relation
3967 @item @code{relation_greater}
3968 @tab @dots{}a @code{>} relation
3969 @item @code{relation_greater_or_equal}
3970 @tab @dots{}a @code{>=} relation
3972 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3974 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3975 @item @code{polynomial}
3976 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3977 @item @code{integer_polynomial}
3978 @tab @dots{}a polynomial with (non-complex) integer coefficients
3979 @item @code{cinteger_polynomial}
3980 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3981 @item @code{rational_polynomial}
3982 @tab @dots{}a polynomial with (non-complex) rational coefficients
3983 @item @code{crational_polynomial}
3984 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3985 @item @code{rational_function}
3986 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3990 To determine whether an expression is commutative or non-commutative and if
3991 so, with which other expressions it would commutate, you use the methods
3992 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3993 for an explanation of these.
3996 @subsection Accessing subexpressions
3999 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
4000 @code{function}, act as containers for subexpressions. For example, the
4001 subexpressions of a sum (an @code{add} object) are the individual terms,
4002 and the subexpressions of a @code{function} are the function's arguments.
4004 @cindex @code{nops()}
4006 GiNaC provides several ways of accessing subexpressions. The first way is to
4011 ex ex::op(size_t i);
4014 @code{nops()} determines the number of subexpressions (operands) contained
4015 in the expression, while @code{op(i)} returns the @code{i}-th
4016 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
4017 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
4018 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
4019 @math{i>0} are the indices.
4022 @cindex @code{const_iterator}
4023 The second way to access subexpressions is via the STL-style random-access
4024 iterator class @code{const_iterator} and the methods
4027 const_iterator ex::begin();
4028 const_iterator ex::end();
4031 @code{begin()} returns an iterator referring to the first subexpression;
4032 @code{end()} returns an iterator which is one-past the last subexpression.
4033 If the expression has no subexpressions, then @code{begin() == end()}. These
4034 iterators can also be used in conjunction with non-modifying STL algorithms.
4036 Here is an example that (non-recursively) prints the subexpressions of a
4037 given expression in three different ways:
4044 for (size_t i = 0; i != e.nops(); ++i)
4045 cout << e.op(i) << endl;
4048 for (const_iterator i = e.begin(); i != e.end(); ++i)
4051 // with iterators and STL copy()
4052 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4056 @cindex @code{const_preorder_iterator}
4057 @cindex @code{const_postorder_iterator}
4058 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4059 expression's immediate children. GiNaC provides two additional iterator
4060 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4061 that iterate over all objects in an expression tree, in preorder or postorder,
4062 respectively. They are STL-style forward iterators, and are created with the
4066 const_preorder_iterator ex::preorder_begin();
4067 const_preorder_iterator ex::preorder_end();
4068 const_postorder_iterator ex::postorder_begin();
4069 const_postorder_iterator ex::postorder_end();
4072 The following example illustrates the differences between
4073 @code{const_iterator}, @code{const_preorder_iterator}, and
4074 @code{const_postorder_iterator}:
4078 symbol A("A"), B("B"), C("C");
4079 ex e = lst@{lst@{A, B@}, C@};
4081 std::copy(e.begin(), e.end(),
4082 std::ostream_iterator<ex>(cout, "\n"));
4086 std::copy(e.preorder_begin(), e.preorder_end(),
4087 std::ostream_iterator<ex>(cout, "\n"));
4094 std::copy(e.postorder_begin(), e.postorder_end(),
4095 std::ostream_iterator<ex>(cout, "\n"));
4104 @cindex @code{relational} (class)
4105 Finally, the left-hand side and right-hand side expressions of objects of
4106 class @code{relational} (and only of these) can also be accessed with the
4115 @subsection Comparing expressions
4116 @cindex @code{is_equal()}
4117 @cindex @code{is_zero()}
4119 Expressions can be compared with the usual C++ relational operators like
4120 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4121 the result is usually not determinable and the result will be @code{false},
4122 except in the case of the @code{!=} operator. You should also be aware that
4123 GiNaC will only do the most trivial test for equality (subtracting both
4124 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4127 Actually, if you construct an expression like @code{a == b}, this will be
4128 represented by an object of the @code{relational} class (@pxref{Relations})
4129 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4131 There are also two methods
4134 bool ex::is_equal(const ex & other);
4138 for checking whether one expression is equal to another, or equal to zero,
4139 respectively. See also the method @code{ex::is_zero_matrix()},
4143 @subsection Ordering expressions
4144 @cindex @code{ex_is_less} (class)
4145 @cindex @code{ex_is_equal} (class)
4146 @cindex @code{compare()}
4148 Sometimes it is necessary to establish a mathematically well-defined ordering
4149 on a set of arbitrary expressions, for example to use expressions as keys
4150 in a @code{std::map<>} container, or to bring a vector of expressions into
4151 a canonical order (which is done internally by GiNaC for sums and products).
4153 The operators @code{<}, @code{>} etc. described in the last section cannot
4154 be used for this, as they don't implement an ordering relation in the
4155 mathematical sense. In particular, they are not guaranteed to be
4156 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4157 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4160 By default, STL classes and algorithms use the @code{<} and @code{==}
4161 operators to compare objects, which are unsuitable for expressions, but GiNaC
4162 provides two functors that can be supplied as proper binary comparison
4163 predicates to the STL:
4168 bool operator()(const ex &lh, const ex &rh) const;
4171 class ex_is_equal @{
4173 bool operator()(const ex &lh, const ex &rh) const;
4177 For example, to define a @code{map} that maps expressions to strings you
4181 std::map<ex, std::string, ex_is_less> myMap;
4184 Omitting the @code{ex_is_less} template parameter will introduce spurious
4185 bugs because the map operates improperly.
4187 Other examples for the use of the functors:
4195 std::sort(v.begin(), v.end(), ex_is_less());
4197 // count the number of expressions equal to '1'
4198 unsigned num_ones = std::count_if(v.begin(), v.end(),
4199 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4202 The implementation of @code{ex_is_less} uses the member function
4205 int ex::compare(const ex & other) const;
4208 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4209 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4213 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4214 @c node-name, next, previous, up
4215 @section Numerical evaluation
4216 @cindex @code{evalf()}
4218 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4219 To evaluate them using floating-point arithmetic you need to call
4222 ex ex::evalf() const;
4225 @cindex @code{Digits}
4226 The accuracy of the evaluation is controlled by the global object @code{Digits}
4227 which can be assigned an integer value. The default value of @code{Digits}
4228 is 17. @xref{Numbers}, for more information and examples.
4230 To evaluate an expression to a @code{double} floating-point number you can
4231 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4235 // Approximate sin(x/Pi)
4237 ex e = series(sin(x/Pi), x == 0, 6);
4239 // Evaluate numerically at x=0.1
4240 ex f = evalf(e.subs(x == 0.1));
4242 // ex_to<numeric> is an unsafe cast, so check the type first
4243 if (is_a<numeric>(f)) @{
4244 double d = ex_to<numeric>(f).to_double();
4253 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4254 @c node-name, next, previous, up
4255 @section Substituting expressions
4256 @cindex @code{subs()}
4258 Algebraic objects inside expressions can be replaced with arbitrary
4259 expressions via the @code{.subs()} method:
4262 ex ex::subs(const ex & e, unsigned options = 0);
4263 ex ex::subs(const exmap & m, unsigned options = 0);
4264 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4267 In the first form, @code{subs()} accepts a relational of the form
4268 @samp{object == expression} or a @code{lst} of such relationals:
4272 symbol x("x"), y("y");
4274 ex e1 = 2*x*x-4*x+3;
4275 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4279 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4284 If you specify multiple substitutions, they are performed in parallel, so e.g.
4285 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4287 The second form of @code{subs()} takes an @code{exmap} object which is a
4288 pair associative container that maps expressions to expressions (currently
4289 implemented as a @code{std::map}). This is the most efficient one of the
4290 three @code{subs()} forms and should be used when the number of objects to
4291 be substituted is large or unknown.
4293 Using this form, the second example from above would look like this:
4297 symbol x("x"), y("y");
4303 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4307 The third form of @code{subs()} takes two lists, one for the objects to be
4308 replaced and one for the expressions to be substituted (both lists must
4309 contain the same number of elements). Using this form, you would write
4313 symbol x("x"), y("y");
4316 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4320 The optional last argument to @code{subs()} is a combination of
4321 @code{subs_options} flags. There are three options available:
4322 @code{subs_options::no_pattern} disables pattern matching, which makes
4323 large @code{subs()} operations significantly faster if you are not using
4324 patterns. The second option, @code{subs_options::algebraic} enables
4325 algebraic substitutions in products and powers.
4326 @xref{Pattern matching and advanced substitutions}, for more information
4327 about patterns and algebraic substitutions. The third option,
4328 @code{subs_options::no_index_renaming} disables the feature that dummy
4329 indices are renamed if the substitution could give a result in which a
4330 dummy index occurs more than two times. This is sometimes necessary if
4331 you want to use @code{subs()} to rename your dummy indices.
4333 @code{subs()} performs syntactic substitution of any complete algebraic
4334 object; it does not try to match sub-expressions as is demonstrated by the
4339 symbol x("x"), y("y"), z("z");
4341 ex e1 = pow(x+y, 2);
4342 cout << e1.subs(x+y == 4) << endl;
4345 ex e2 = sin(x)*sin(y)*cos(x);
4346 cout << e2.subs(sin(x) == cos(x)) << endl;
4347 // -> cos(x)^2*sin(y)
4350 cout << e3.subs(x+y == 4) << endl;
4352 // (and not 4+z as one might expect)
4356 A more powerful form of substitution using wildcards is described in the
4360 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4361 @c node-name, next, previous, up
4362 @section Pattern matching and advanced substitutions
4363 @cindex @code{wildcard} (class)
4364 @cindex Pattern matching
4366 GiNaC allows the use of patterns for checking whether an expression is of a
4367 certain form or contains subexpressions of a certain form, and for
4368 substituting expressions in a more general way.
4370 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4371 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4372 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4373 an unsigned integer number to allow having multiple different wildcards in a
4374 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4375 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4379 ex wild(unsigned label = 0);
4382 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4385 Some examples for patterns:
4387 @multitable @columnfractions .5 .5
4388 @item @strong{Constructed as} @tab @strong{Output as}
4389 @item @code{wild()} @tab @samp{$0}
4390 @item @code{pow(x,wild())} @tab @samp{x^$0}
4391 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4392 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4398 @item Wildcards behave like symbols and are subject to the same algebraic
4399 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4400 @item As shown in the last example, to use wildcards for indices you have to
4401 use them as the value of an @code{idx} object. This is because indices must
4402 always be of class @code{idx} (or a subclass).
4403 @item Wildcards only represent expressions or subexpressions. It is not
4404 possible to use them as placeholders for other properties like index
4405 dimension or variance, representation labels, symmetry of indexed objects
4407 @item Because wildcards are commutative, it is not possible to use wildcards
4408 as part of noncommutative products.
4409 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4410 are also valid patterns.
4413 @subsection Matching expressions
4414 @cindex @code{match()}
4415 The most basic application of patterns is to check whether an expression
4416 matches a given pattern. This is done by the function
4419 bool ex::match(const ex & pattern);
4420 bool ex::match(const ex & pattern, exmap& repls);
4423 This function returns @code{true} when the expression matches the pattern
4424 and @code{false} if it doesn't. If used in the second form, the actual
4425 subexpressions matched by the wildcards get returned in the associative
4426 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4427 returns false, @code{repls} remains unmodified.
4429 The matching algorithm works as follows:
4432 @item A single wildcard matches any expression. If one wildcard appears
4433 multiple times in a pattern, it must match the same expression in all
4434 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4435 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4436 @item If the expression is not of the same class as the pattern, the match
4437 fails (i.e. a sum only matches a sum, a function only matches a function,
4439 @item If the pattern is a function, it only matches the same function
4440 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4441 @item Except for sums and products, the match fails if the number of
4442 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4444 @item If there are no subexpressions, the expressions and the pattern must
4445 be equal (in the sense of @code{is_equal()}).
4446 @item Except for sums and products, each subexpression (@code{op()}) must
4447 match the corresponding subexpression of the pattern.
4450 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4451 account for their commutativity and associativity:
4454 @item If the pattern contains a term or factor that is a single wildcard,
4455 this one is used as the @dfn{global wildcard}. If there is more than one
4456 such wildcard, one of them is chosen as the global wildcard in a random
4458 @item Every term/factor of the pattern, except the global wildcard, is
4459 matched against every term of the expression in sequence. If no match is
4460 found, the whole match fails. Terms that did match are not considered in
4462 @item If there are no unmatched terms left, the match succeeds. Otherwise
4463 the match fails unless there is a global wildcard in the pattern, in
4464 which case this wildcard matches the remaining terms.
4467 In general, having more than one single wildcard as a term of a sum or a
4468 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4471 Here are some examples in @command{ginsh} to demonstrate how it works (the
4472 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4473 match fails, and the list of wildcard replacements otherwise):
4476 > match((x+y)^a,(x+y)^a);
4478 > match((x+y)^a,(x+y)^b);
4480 > match((x+y)^a,$1^$2);
4482 > match((x+y)^a,$1^$1);
4484 > match((x+y)^(x+y),$1^$1);
4486 > match((x+y)^(x+y),$1^$2);
4488 > match((a+b)*(a+c),($1+b)*($1+c));
4490 > match((a+b)*(a+c),(a+$1)*(a+$2));
4492 (Unpredictable. The result might also be [$1==c,$2==b].)
4493 > match((a+b)*(a+c),($1+$2)*($1+$3));
4494 (The result is undefined. Due to the sequential nature of the algorithm
4495 and the re-ordering of terms in GiNaC, the match for the first factor
4496 may be @{$1==a,$2==b@} in which case the match for the second factor
4497 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4499 > match(a*(x+y)+a*z+b,a*$1+$2);
4500 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4501 @{$1=x+y,$2=a*z+b@}.)
4502 > match(a+b+c+d+e+f,c);
4504 > match(a+b+c+d+e+f,c+$0);
4506 > match(a+b+c+d+e+f,c+e+$0);
4508 > match(a+b,a+b+$0);
4510 > match(a*b^2,a^$1*b^$2);
4512 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4513 even though a==a^1.)
4514 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4516 > match(atan2(y,x^2),atan2(y,$0));
4520 @subsection Matching parts of expressions
4521 @cindex @code{has()}
4522 A more general way to look for patterns in expressions is provided by the
4526 bool ex::has(const ex & pattern);
4529 This function checks whether a pattern is matched by an expression itself or
4530 by any of its subexpressions.
4532 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4533 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4536 > has(x*sin(x+y+2*a),y);
4538 > has(x*sin(x+y+2*a),x+y);
4540 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4541 has the subexpressions "x", "y" and "2*a".)
4542 > has(x*sin(x+y+2*a),x+y+$1);
4544 (But this is possible.)
4545 > has(x*sin(2*(x+y)+2*a),x+y);
4547 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4548 which "x+y" is not a subexpression.)
4551 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4553 > has(4*x^2-x+3,$1*x);
4555 > has(4*x^2+x+3,$1*x);
4557 (Another possible pitfall. The first expression matches because the term
4558 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4559 contains a linear term you should use the coeff() function instead.)
4562 @cindex @code{find()}
4566 bool ex::find(const ex & pattern, exset& found);
4569 works a bit like @code{has()} but it doesn't stop upon finding the first
4570 match. Instead, it appends all found matches to the specified list. If there
4571 are multiple occurrences of the same expression, it is entered only once to
4572 the list. @code{find()} returns false if no matches were found (in
4573 @command{ginsh}, it returns an empty list):
4576 > find(1+x+x^2+x^3,x);
4578 > find(1+x+x^2+x^3,y);
4580 > find(1+x+x^2+x^3,x^$1);
4582 (Note the absence of "x".)
4583 > expand((sin(x)+sin(y))*(a+b));
4584 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4589 @subsection Substituting expressions
4590 @cindex @code{subs()}
4591 Probably the most useful application of patterns is to use them for
4592 substituting expressions with the @code{subs()} method. Wildcards can be
4593 used in the search patterns as well as in the replacement expressions, where
4594 they get replaced by the expressions matched by them. @code{subs()} doesn't
4595 know anything about algebra; it performs purely syntactic substitutions.
4600 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4602 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4604 > subs((a+b+c)^2,a+b==x);
4606 > subs((a+b+c)^2,a+b+$1==x+$1);
4608 > subs(a+2*b,a+b==x);
4610 > subs(4*x^3-2*x^2+5*x-1,x==a);
4612 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4614 > subs(sin(1+sin(x)),sin($1)==cos($1));
4616 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4620 The last example would be written in C++ in this way:
4624 symbol a("a"), b("b"), x("x"), y("y");
4625 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4626 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4627 cout << e.expand() << endl;
4632 @subsection The option algebraic
4633 Both @code{has()} and @code{subs()} take an optional argument to pass them
4634 extra options. This section describes what happens if you give the former
4635 the option @code{has_options::algebraic} or the latter
4636 @code{subs_options::algebraic}. In that case the matching condition for
4637 powers and multiplications is changed in such a way that they become
4638 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4639 If you use these options you will find that
4640 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4641 Besides matching some of the factors of a product also powers match as
4642 often as is possible without getting negative exponents. For example
4643 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4644 @code{x*c^2*z}. This also works with negative powers:
4645 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4646 return @code{x^(-1)*c^2*z}.
4648 @strong{Please notice:} this only works for multiplications
4649 and not for locating @code{x+y} within @code{x+y+z}.
4652 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4653 @c node-name, next, previous, up
4654 @section Applying a function on subexpressions
4655 @cindex tree traversal
4656 @cindex @code{map()}
4658 Sometimes you may want to perform an operation on specific parts of an
4659 expression while leaving the general structure of it intact. An example
4660 of this would be a matrix trace operation: the trace of a sum is the sum
4661 of the traces of the individual terms. That is, the trace should @dfn{map}
4662 on the sum, by applying itself to each of the sum's operands. It is possible
4663 to do this manually which usually results in code like this:
4668 if (is_a<matrix>(e))
4669 return ex_to<matrix>(e).trace();
4670 else if (is_a<add>(e)) @{
4672 for (size_t i=0; i<e.nops(); i++)
4673 sum += calc_trace(e.op(i));
4675 @} else if (is_a<mul>)(e)) @{
4683 This is, however, slightly inefficient (if the sum is very large it can take
4684 a long time to add the terms one-by-one), and its applicability is limited to
4685 a rather small class of expressions. If @code{calc_trace()} is called with
4686 a relation or a list as its argument, you will probably want the trace to
4687 be taken on both sides of the relation or of all elements of the list.
4689 GiNaC offers the @code{map()} method to aid in the implementation of such
4693 ex ex::map(map_function & f) const;
4694 ex ex::map(ex (*f)(const ex & e)) const;
4697 In the first (preferred) form, @code{map()} takes a function object that
4698 is subclassed from the @code{map_function} class. In the second form, it
4699 takes a pointer to a function that accepts and returns an expression.
4700 @code{map()} constructs a new expression of the same type, applying the
4701 specified function on all subexpressions (in the sense of @code{op()}),
4704 The use of a function object makes it possible to supply more arguments to
4705 the function that is being mapped, or to keep local state information.
4706 The @code{map_function} class declares a virtual function call operator
4707 that you can overload. Here is a sample implementation of @code{calc_trace()}
4708 that uses @code{map()} in a recursive fashion:
4711 struct calc_trace : public map_function @{
4712 ex operator()(const ex &e)
4714 if (is_a<matrix>(e))
4715 return ex_to<matrix>(e).trace();
4716 else if (is_a<mul>(e)) @{
4719 return e.map(*this);
4724 This function object could then be used like this:
4728 ex M = ... // expression with matrices
4729 calc_trace do_trace;
4730 ex tr = do_trace(M);
4734 Here is another example for you to meditate over. It removes quadratic
4735 terms in a variable from an expanded polynomial:
4738 struct map_rem_quad : public map_function @{
4740 map_rem_quad(const ex & var_) : var(var_) @{@}
4742 ex operator()(const ex & e)
4744 if (is_a<add>(e) || is_a<mul>(e))
4745 return e.map(*this);
4746 else if (is_a<power>(e) &&
4747 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4757 symbol x("x"), y("y");
4760 for (int i=0; i<8; i++)
4761 e += pow(x, i) * pow(y, 8-i) * (i+1);
4763 // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
4765 map_rem_quad rem_quad(x);
4766 cout << rem_quad(e) << endl;
4767 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4771 @command{ginsh} offers a slightly different implementation of @code{map()}
4772 that allows applying algebraic functions to operands. The second argument
4773 to @code{map()} is an expression containing the wildcard @samp{$0} which
4774 acts as the placeholder for the operands:
4779 > map(a+2*b,sin($0));
4781 > map(@{a,b,c@},$0^2+$0);
4782 @{a^2+a,b^2+b,c^2+c@}
4785 Note that it is only possible to use algebraic functions in the second
4786 argument. You can not use functions like @samp{diff()}, @samp{op()},
4787 @samp{subs()} etc. because these are evaluated immediately:
4790 > map(@{a,b,c@},diff($0,a));
4792 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4793 to "map(@{a,b,c@},0)".
4797 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4798 @c node-name, next, previous, up
4799 @section Visitors and tree traversal
4800 @cindex tree traversal
4801 @cindex @code{visitor} (class)
4802 @cindex @code{accept()}
4803 @cindex @code{visit()}
4804 @cindex @code{traverse()}
4805 @cindex @code{traverse_preorder()}
4806 @cindex @code{traverse_postorder()}
4808 Suppose that you need a function that returns a list of all indices appearing
4809 in an arbitrary expression. The indices can have any dimension, and for
4810 indices with variance you always want the covariant version returned.
4812 You can't use @code{get_free_indices()} because you also want to include
4813 dummy indices in the list, and you can't use @code{find()} as it needs
4814 specific index dimensions (and it would require two passes: one for indices
4815 with variance, one for plain ones).
4817 The obvious solution to this problem is a tree traversal with a type switch,
4818 such as the following:
4821 void gather_indices_helper(const ex & e, lst & l)
4823 if (is_a<varidx>(e)) @{
4824 const varidx & vi = ex_to<varidx>(e);
4825 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4826 @} else if (is_a<idx>(e)) @{
4829 size_t n = e.nops();
4830 for (size_t i = 0; i < n; ++i)
4831 gather_indices_helper(e.op(i), l);
4835 lst gather_indices(const ex & e)
4838 gather_indices_helper(e, l);
4845 This works fine but fans of object-oriented programming will feel
4846 uncomfortable with the type switch. One reason is that there is a possibility
4847 for subtle bugs regarding derived classes. If we had, for example, written
4850 if (is_a<idx>(e)) @{
4852 @} else if (is_a<varidx>(e)) @{
4856 in @code{gather_indices_helper}, the code wouldn't have worked because the
4857 first line "absorbs" all classes derived from @code{idx}, including
4858 @code{varidx}, so the special case for @code{varidx} would never have been
4861 Also, for a large number of classes, a type switch like the above can get
4862 unwieldy and inefficient (it's a linear search, after all).
4863 @code{gather_indices_helper} only checks for two classes, but if you had to
4864 write a function that required a different implementation for nearly
4865 every GiNaC class, the result would be very hard to maintain and extend.
4867 The cleanest approach to the problem would be to add a new virtual function
4868 to GiNaC's class hierarchy. In our example, there would be specializations
4869 for @code{idx} and @code{varidx} while the default implementation in
4870 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4871 impossible to add virtual member functions to existing classes without
4872 changing their source and recompiling everything. GiNaC comes with source,
4873 so you could actually do this, but for a small algorithm like the one
4874 presented this would be impractical.
4876 One solution to this dilemma is the @dfn{Visitor} design pattern,
4877 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4878 variation, described in detail in
4879 @uref{https://condor.depaul.edu/dmumaugh/OOT/Design-Principles/acv.pdf}). Instead of adding
4880 virtual functions to the class hierarchy to implement operations, GiNaC
4881 provides a single "bouncing" method @code{accept()} that takes an instance
4882 of a special @code{visitor} class and redirects execution to the one
4883 @code{visit()} virtual function of the visitor that matches the type of
4884 object that @code{accept()} was being invoked on.
4886 Visitors in GiNaC must derive from the global @code{visitor} class as well
4887 as from the class @code{T::visitor} of each class @code{T} they want to
4888 visit, and implement the member functions @code{void visit(const T &)} for
4894 void ex::accept(visitor & v) const;
4897 will then dispatch to the correct @code{visit()} member function of the
4898 specified visitor @code{v} for the type of GiNaC object at the root of the
4899 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4901 Here is an example of a visitor:
4905 : public visitor, // this is required
4906 public add::visitor, // visit add objects
4907 public numeric::visitor, // visit numeric objects
4908 public basic::visitor // visit basic objects
4910 void visit(const add & x)
4911 @{ cout << "called with an add object" << endl; @}
4913 void visit(const numeric & x)
4914 @{ cout << "called with a numeric object" << endl; @}
4916 void visit(const basic & x)
4917 @{ cout << "called with a basic object" << endl; @}
4921 which can be used as follows:
4932 // prints "called with a numeric object"
4934 // prints "called with an add object"
4936 // prints "called with a basic object"
4940 The @code{visit(const basic &)} method gets called for all objects that are
4941 not @code{numeric} or @code{add} and acts as an (optional) default.
4943 From a conceptual point of view, the @code{visit()} methods of the visitor
4944 behave like a newly added virtual function of the visited hierarchy.
4945 In addition, visitors can store state in member variables, and they can
4946 be extended by deriving a new visitor from an existing one, thus building
4947 hierarchies of visitors.
4949 We can now rewrite our index example from above with a visitor:
4952 class gather_indices_visitor
4953 : public visitor, public idx::visitor, public varidx::visitor
4957 void visit(const idx & i)
4962 void visit(const varidx & vi)
4964 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4968 const lst & get_result() // utility function
4977 What's missing is the tree traversal. We could implement it in
4978 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4981 void ex::traverse_preorder(visitor & v) const;
4982 void ex::traverse_postorder(visitor & v) const;
4983 void ex::traverse(visitor & v) const;
4986 @code{traverse_preorder()} visits a node @emph{before} visiting its
4987 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4988 visiting its subexpressions. @code{traverse()} is a synonym for
4989 @code{traverse_preorder()}.
4991 Here is a new implementation of @code{gather_indices()} that uses the visitor
4992 and @code{traverse()}:
4995 lst gather_indices(const ex & e)
4997 gather_indices_visitor v;
4999 return v.get_result();
5003 Alternatively, you could use pre- or postorder iterators for the tree
5007 lst gather_indices(const ex & e)
5009 gather_indices_visitor v;
5010 for (const_preorder_iterator i = e.preorder_begin();
5011 i != e.preorder_end(); ++i) @{
5014 return v.get_result();
5019 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
5020 @c node-name, next, previous, up
5021 @section Polynomial arithmetic
5023 @subsection Testing whether an expression is a polynomial
5024 @cindex @code{is_polynomial()}
5026 Testing whether an expression is a polynomial in one or more variables
5027 can be done with the method
5029 bool ex::is_polynomial(const ex & vars) const;
5031 In the case of more than
5032 one variable, the variables are given as a list.
5035 (x*y*sin(y)).is_polynomial(x) // Returns true.
5036 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5039 @subsection Expanding and collecting
5040 @cindex @code{expand()}
5041 @cindex @code{collect()}
5042 @cindex @code{collect_common_factors()}
5044 A polynomial in one or more variables has many equivalent
5045 representations. Some useful ones serve a specific purpose. Consider
5046 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5047 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5048 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5049 representations are the recursive ones where one collects for exponents
5050 in one of the three variable. Since the factors are themselves
5051 polynomials in the remaining two variables the procedure can be
5052 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5053 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5056 To bring an expression into expanded form, its method
5059 ex ex::expand(unsigned options = 0);
5062 may be called. In our example above, this corresponds to @math{4*x*y +
5063 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5064 GiNaC is not easy to guess you should be prepared to see different
5065 orderings of terms in such sums!
5067 Another useful representation of multivariate polynomials is as a
5068 univariate polynomial in one of the variables with the coefficients
5069 being polynomials in the remaining variables. The method
5070 @code{collect()} accomplishes this task:
5073 ex ex::collect(const ex & s, bool distributed = false);
5076 The first argument to @code{collect()} can also be a list of objects in which
5077 case the result is either a recursively collected polynomial, or a polynomial
5078 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5079 by the @code{distributed} flag.
5081 Note that the original polynomial needs to be in expanded form (for the
5082 variables concerned) in order for @code{collect()} to be able to find the
5083 coefficients properly.
5085 The following @command{ginsh} transcript shows an application of @code{collect()}
5086 together with @code{find()}:
5089 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5090 d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)
5091 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5092 > collect(a,@{p,q@});
5093 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5094 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5095 > collect(a,find(a,sin($1)));
5096 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5097 > collect(a,@{find(a,sin($1)),p,q@});
5098 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5099 > collect(a,@{find(a,sin($1)),d@});
5100 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5103 Polynomials can often be brought into a more compact form by collecting
5104 common factors from the terms of sums. This is accomplished by the function
5107 ex collect_common_factors(const ex & e);
5110 This function doesn't perform a full factorization but only looks for
5111 factors which are already explicitly present:
5114 > collect_common_factors(a*x+a*y);
5116 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5118 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5119 (c+a)*a*(x*y+y^2+x)*b
5122 @subsection Degree and coefficients
5123 @cindex @code{degree()}
5124 @cindex @code{ldegree()}
5125 @cindex @code{coeff()}
5127 The degree and low degree of a polynomial in expanded form can be obtained
5128 using the two methods
5131 int ex::degree(const ex & s);
5132 int ex::ldegree(const ex & s);
5135 These functions even work on rational functions, returning the asymptotic
5136 degree. By definition, the degree of zero is zero. To extract a coefficient
5137 with a certain power from an expanded polynomial you use
5140 ex ex::coeff(const ex & s, int n);
5143 You can also obtain the leading and trailing coefficients with the methods
5146 ex ex::lcoeff(const ex & s);
5147 ex ex::tcoeff(const ex & s);
5150 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5153 An application is illustrated in the next example, where a multivariate
5154 polynomial is analyzed:
5158 symbol x("x"), y("y");
5159 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5160 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5161 ex Poly = PolyInp.expand();
5163 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5164 cout << "The x^" << i << "-coefficient is "
5165 << Poly.coeff(x,i) << endl;
5167 cout << "As polynomial in y: "
5168 << Poly.collect(y) << endl;
5172 When run, it returns an output in the following fashion:
5175 The x^0-coefficient is y^2+11*y
5176 The x^1-coefficient is 5*y^2-2*y
5177 The x^2-coefficient is -1
5178 The x^3-coefficient is 4*y
5179 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5182 As always, the exact output may vary between different versions of GiNaC
5183 or even from run to run since the internal canonical ordering is not
5184 within the user's sphere of influence.
5186 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5187 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5188 with non-polynomial expressions as they not only work with symbols but with
5189 constants, functions and indexed objects as well:
5193 symbol a("a"), b("b"), c("c"), x("x");
5194 idx i(symbol("i"), 3);
5196 ex e = pow(sin(x) - cos(x), 4);
5197 cout << e.degree(cos(x)) << endl;
5199 cout << e.expand().coeff(sin(x), 3) << endl;
5202 e = indexed(a+b, i) * indexed(b+c, i);
5203 e = e.expand(expand_options::expand_indexed);
5204 cout << e.collect(indexed(b, i)) << endl;
5205 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5210 @subsection Polynomial division
5211 @cindex polynomial division
5214 @cindex pseudo-remainder
5215 @cindex @code{quo()}
5216 @cindex @code{rem()}
5217 @cindex @code{prem()}
5218 @cindex @code{divide()}
5223 ex quo(const ex & a, const ex & b, const ex & x);
5224 ex rem(const ex & a, const ex & b, const ex & x);
5227 compute the quotient and remainder of univariate polynomials in the variable
5228 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5230 The additional function
5233 ex prem(const ex & a, const ex & b, const ex & x);
5236 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5237 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5239 Exact division of multivariate polynomials is performed by the function
5242 bool divide(const ex & a, const ex & b, ex & q);
5245 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5246 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5247 in which case the value of @code{q} is undefined.
5250 @subsection Unit, content and primitive part
5251 @cindex @code{unit()}
5252 @cindex @code{content()}
5253 @cindex @code{primpart()}
5254 @cindex @code{unitcontprim()}
5259 ex ex::unit(const ex & x);
5260 ex ex::content(const ex & x);
5261 ex ex::primpart(const ex & x);
5262 ex ex::primpart(const ex & x, const ex & c);
5265 return the unit part, content part, and primitive polynomial of a multivariate
5266 polynomial with respect to the variable @samp{x} (the unit part being the sign
5267 of the leading coefficient, the content part being the GCD of the coefficients,
5268 and the primitive polynomial being the input polynomial divided by the unit and
5269 content parts). The second variant of @code{primpart()} expects the previously
5270 calculated content part of the polynomial in @code{c}, which enables it to
5271 work faster in the case where the content part has already been computed. The
5272 product of unit, content, and primitive part is the original polynomial.
5274 Additionally, the method
5277 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5280 computes the unit, content, and primitive parts in one go, returning them
5281 in @code{u}, @code{c}, and @code{p}, respectively.
5284 @subsection GCD, LCM and resultant
5287 @cindex @code{gcd()}
5288 @cindex @code{lcm()}
5290 The functions for polynomial greatest common divisor and least common
5291 multiple have the synopsis
5294 ex gcd(const ex & a, const ex & b);
5295 ex lcm(const ex & a, const ex & b);
5298 The functions @code{gcd()} and @code{lcm()} accept two expressions
5299 @code{a} and @code{b} as arguments and return a new expression, their
5300 greatest common divisor or least common multiple, respectively. If the
5301 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5302 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5303 the coefficients must be rationals.
5306 #include <ginac/ginac.h>
5307 using namespace GiNaC;
5311 symbol x("x"), y("y"), z("z");
5312 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5313 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5315 ex P_gcd = gcd(P_a, P_b);
5317 ex P_lcm = lcm(P_a, P_b);
5318 // 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
5323 @cindex @code{resultant()}
5325 The resultant of two expressions only makes sense with polynomials.
5326 It is always computed with respect to a specific symbol within the
5327 expressions. The function has the interface
5330 ex resultant(const ex & a, const ex & b, const ex & s);
5333 Resultants are symmetric in @code{a} and @code{b}. The following example
5334 computes the resultant of two expressions with respect to @code{x} and
5335 @code{y}, respectively:
5338 #include <ginac/ginac.h>
5339 using namespace GiNaC;
5343 symbol x("x"), y("y");
5345 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5348 r = resultant(e1, e2, x);
5350 r = resultant(e1, e2, y);
5355 @subsection Square-free decomposition
5356 @cindex square-free decomposition
5357 @cindex factorization
5358 @cindex @code{sqrfree()}
5360 Square-free decomposition is available in GiNaC:
5362 ex sqrfree(const ex & a, const lst & l = lst@{@});
5364 Here is an example that by the way illustrates how the exact form of the
5365 result may slightly depend on the order of differentiation, calling for
5366 some care with subsequent processing of the result:
5369 symbol x("x"), y("y");
5370 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5372 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5373 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5375 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5376 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5378 cout << sqrfree(BiVarPol) << endl;
5379 // -> depending on luck, any of the above
5382 Note also, how factors with the same exponents are not fully factorized
5385 @subsection Square-free partial fraction decomposition
5386 @cindex square-free partial fraction decomposition
5387 @cindex partial fraction decomposition
5388 @cindex @code{sqrfree_parfrac()}
5390 GiNaC also supports square-free partial fraction decomposition of
5393 ex sqrfree_parfrac(const ex & a, const symbol & x);
5395 It is called square-free because it assumes a square-free
5396 factorization of the input's denominator:
5401 ex rat = (x-4)/(pow(x,2)*(x+2));
5402 cout << sqrfree_parfrac(rat, x) << endl;
5403 // -> -2*x^(-2)+3/2*x^(-1)-3/2*(2+x)^(-1)
5406 @subsection Polynomial factorization
5407 @cindex factorization
5408 @cindex polynomial factorization
5409 @cindex @code{factor()}
5411 Polynomials can also be fully factored with a call to the function
5413 ex factor(const ex & a, unsigned int options = 0);
5415 The factorization works for univariate and multivariate polynomials with
5416 rational coefficients. The following code snippet shows its capabilities:
5419 cout << factor(pow(x,2)-1) << endl;
5421 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5422 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5423 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5424 // -> -1+sin(-1+x^2)+x^2
5427 The results are as expected except for the last one where no factorization
5428 seems to have been done. This is due to the default option
5429 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5430 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5431 In the shown example this is not the case, because one term is a function.
5433 There exists a second option @command{factor_options::all}, which tells GiNaC to
5434 ignore non-polynomial parts of an expression and also to look inside function
5435 arguments. With this option the example gives:
5438 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5440 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5443 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5444 the following example does not factor:
5447 cout << factor(pow(x,2)-2) << endl;
5448 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5451 Factorization is useful in many applications. A lot of algorithms in computer
5452 algebra depend on the ability to factor a polynomial. Of course, factorization
5453 can also be used to simplify expressions, but it is costly and applying it to
5454 complicated expressions (high degrees or many terms) may consume far too much
5455 time. So usually, looking for a GCD at strategic points in a calculation is the
5456 cheaper and more appropriate alternative.
5458 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5459 @c node-name, next, previous, up
5460 @section Rational expressions
5462 @subsection The @code{normal} method
5463 @cindex @code{normal()}
5464 @cindex simplification
5465 @cindex temporary replacement
5467 Some basic form of simplification of expressions is called for frequently.
5468 GiNaC provides the method @code{.normal()}, which converts a rational function
5469 into an equivalent rational function of the form @samp{numerator/denominator}
5470 where numerator and denominator are coprime. If the input expression is already
5471 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5472 otherwise it performs fraction addition and multiplication.
5474 @code{.normal()} can also be used on expressions which are not rational functions
5475 as it will replace all non-rational objects (like functions or non-integer
5476 powers) by temporary symbols to bring the expression to the domain of rational
5477 functions before performing the normalization, and re-substituting these
5478 symbols afterwards. This algorithm is also available as a separate method
5479 @code{.to_rational()}, described below.
5481 This means that both expressions @code{t1} and @code{t2} are indeed
5482 simplified in this little code snippet:
5487 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5488 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5489 std::cout << "t1 is " << t1.normal() << std::endl;
5490 std::cout << "t2 is " << t2.normal() << std::endl;
5494 Of course this works for multivariate polynomials too, so the ratio of
5495 the sample-polynomials from the section about GCD and LCM above would be
5496 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5499 @subsection Numerator and denominator
5502 @cindex @code{numer()}
5503 @cindex @code{denom()}
5504 @cindex @code{numer_denom()}
5506 The numerator and denominator of an expression can be obtained with
5511 ex ex::numer_denom();
5514 These functions will first normalize the expression as described above and
5515 then return the numerator, denominator, or both as a list, respectively.
5516 If you need both numerator and denominator, call @code{numer_denom()}: it
5517 is faster than using @code{numer()} and @code{denom()} separately. And even
5518 more important: a separate evaluation of @code{numer()} and @code{denom()}
5519 may result in a spurious sign, e.g. for $x/(x^2-1)$ @code{numer()} may
5520 return $x$ and @code{denom()} $1-x^2$.
5523 @subsection Converting to a polynomial or rational expression
5524 @cindex @code{to_polynomial()}
5525 @cindex @code{to_rational()}
5527 Some of the methods described so far only work on polynomials or rational
5528 functions. GiNaC provides a way to extend the domain of these functions to
5529 general expressions by using the temporary replacement algorithm described
5530 above. You do this by calling
5533 ex ex::to_polynomial(exmap & m);
5537 ex ex::to_rational(exmap & m);
5540 on the expression to be converted. The supplied @code{exmap} will be filled
5541 with the generated temporary symbols and their replacement expressions in a
5542 format that can be used directly for the @code{subs()} method. It can also
5543 already contain a list of replacements from an earlier application of
5544 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5545 it on multiple expressions and get consistent results.
5547 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5548 is probably best illustrated with an example:
5552 symbol x("x"), y("y");
5553 ex a = 2*x/sin(x) - y/(3*sin(x));
5557 ex p = a.to_polynomial(mp);
5558 cout << " = " << p << "\n with " << mp << endl;
5559 // = symbol3*symbol2*y+2*symbol2*x
5560 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5563 ex r = a.to_rational(mr);
5564 cout << " = " << r << "\n with " << mr << endl;
5565 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5566 // with @{symbol4==sin(x)@}
5570 The following more useful example will print @samp{sin(x)-cos(x)}:
5575 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5576 ex b = sin(x) + cos(x);
5579 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5580 cout << q.subs(m) << endl;
5585 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5586 @c node-name, next, previous, up
5587 @section Symbolic differentiation
5588 @cindex differentiation
5589 @cindex @code{diff()}
5591 @cindex product rule
5593 GiNaC's objects know how to differentiate themselves. Thus, a
5594 polynomial (class @code{add}) knows that its derivative is the sum of
5595 the derivatives of all the monomials:
5599 symbol x("x"), y("y"), z("z");
5600 ex P = pow(x, 5) + pow(x, 2) + y;
5602 cout << P.diff(x,2) << endl;
5604 cout << P.diff(y) << endl; // 1
5606 cout << P.diff(z) << endl; // 0
5611 If a second integer parameter @var{n} is given, the @code{diff} method
5612 returns the @var{n}th derivative.
5614 If @emph{every} object and every function is told what its derivative
5615 is, all derivatives of composed objects can be calculated using the
5616 chain rule and the product rule. Consider, for instance the expression
5617 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5618 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5619 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5620 out that the composition is the generating function for Euler Numbers,
5621 i.e. the so called @var{n}th Euler number is the coefficient of
5622 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5623 identity to code a function that generates Euler numbers in just three
5626 @cindex Euler numbers
5628 #include <ginac/ginac.h>
5629 using namespace GiNaC;
5631 ex EulerNumber(unsigned n)
5634 const ex generator = pow(cosh(x),-1);
5635 return generator.diff(x,n).subs(x==0);
5640 for (unsigned i=0; i<11; i+=2)
5641 std::cout << EulerNumber(i) << std::endl;
5646 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5647 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5648 @code{i} by two since all odd Euler numbers vanish anyways.
5651 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5652 @c node-name, next, previous, up
5653 @section Series expansion
5654 @cindex @code{series()}
5655 @cindex Taylor expansion
5656 @cindex Laurent expansion
5657 @cindex @code{pseries} (class)
5658 @cindex @code{Order()}
5660 Expressions know how to expand themselves as a Taylor series or (more
5661 generally) a Laurent series. As in most conventional Computer Algebra
5662 Systems, no distinction is made between those two. There is a class of
5663 its own for storing such series (@code{class pseries}) and a built-in
5664 function (called @code{Order}) for storing the order term of the series.
5665 As a consequence, if you want to work with series, i.e. multiply two
5666 series, you need to call the method @code{ex::series} again to convert
5667 it to a series object with the usual structure (expansion plus order
5668 term). A sample application from special relativity could read:
5671 #include <ginac/ginac.h>
5672 using namespace std;
5673 using namespace GiNaC;
5677 symbol v("v"), c("c");
5679 ex gamma = 1/sqrt(1 - pow(v/c,2));
5680 ex mass_nonrel = gamma.series(v==0, 10);
5682 cout << "the relativistic mass increase with v is " << endl
5683 << mass_nonrel << endl;
5685 cout << "the inverse square of this series is " << endl
5686 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5690 Only calling the series method makes the last output simplify to
5691 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5692 series raised to the power @math{-2}.
5694 @cindex Machin's formula
5695 As another instructive application, let us calculate the numerical
5696 value of Archimedes' constant
5703 (for which there already exists the built-in constant @code{Pi})
5704 using John Machin's amazing formula
5706 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5709 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5711 This equation (and similar ones) were used for over 200 years for
5712 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5713 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5714 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5715 order term with it and the question arises what the system is supposed
5716 to do when the fractions are plugged into that order term. The solution
5717 is to use the function @code{series_to_poly()} to simply strip the order
5721 #include <ginac/ginac.h>
5722 using namespace GiNaC;
5724 ex machin_pi(int degr)
5727 ex pi_expansion = series_to_poly(atan(x).series(x==0,degr));
5728 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5729 -4*pi_expansion.subs(x==numeric(1,239));
5735 using std::cout; // just for fun, another way of...
5736 using std::endl; // ...dealing with this namespace std.
5738 for (int i=2; i<12; i+=2) @{
5739 pi_frac = machin_pi(i);
5740 cout << i << ":\t" << pi_frac << endl
5741 << "\t" << pi_frac.evalf() << endl;
5747 Note how we just called @code{.series(x,degr)} instead of
5748 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5749 method @code{series()}: if the first argument is a symbol the expression
5750 is expanded in that symbol around point @code{0}. When you run this
5751 program, it will type out:
5755 3.1832635983263598326
5756 4: 5359397032/1706489875
5757 3.1405970293260603143
5758 6: 38279241713339684/12184551018734375
5759 3.141621029325034425
5760 8: 76528487109180192540976/24359780855939418203125
5761 3.141591772182177295
5762 10: 327853873402258685803048818236/104359128170408663038552734375
5763 3.1415926824043995174
5767 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5768 @c node-name, next, previous, up
5769 @section Symmetrization
5770 @cindex @code{symmetrize()}
5771 @cindex @code{antisymmetrize()}
5772 @cindex @code{symmetrize_cyclic()}
5777 ex ex::symmetrize(const lst & l);
5778 ex ex::antisymmetrize(const lst & l);
5779 ex ex::symmetrize_cyclic(const lst & l);
5782 symmetrize an expression by returning the sum over all symmetric,
5783 antisymmetric or cyclic permutations of the specified list of objects,
5784 weighted by the number of permutations.
5786 The three additional methods
5789 ex ex::symmetrize();
5790 ex ex::antisymmetrize();
5791 ex ex::symmetrize_cyclic();
5794 symmetrize or antisymmetrize an expression over its free indices.
5796 Symmetrization is most useful with indexed expressions but can be used with
5797 almost any kind of object (anything that is @code{subs()}able):
5801 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5802 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5804 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5805 // -> 1/2*A.j.i+1/2*A.i.j
5806 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5807 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5808 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5809 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5815 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5816 @c node-name, next, previous, up
5817 @section Predefined mathematical functions
5819 @subsection Overview
5821 GiNaC contains the following predefined mathematical functions:
5824 @multitable @columnfractions .30 .70
5825 @item @strong{Name} @tab @strong{Function}
5828 @cindex @code{abs()}
5829 @item @code{step(x)}
5831 @cindex @code{step()}
5832 @item @code{csgn(x)}
5834 @cindex @code{conjugate()}
5835 @item @code{conjugate(x)}
5836 @tab complex conjugation
5837 @cindex @code{real_part()}
5838 @item @code{real_part(x)}
5840 @cindex @code{imag_part()}
5841 @item @code{imag_part(x)}
5843 @item @code{sqrt(x)}
5844 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5845 @cindex @code{sqrt()}
5848 @cindex @code{sin()}
5851 @cindex @code{cos()}
5854 @cindex @code{tan()}
5855 @item @code{asin(x)}
5857 @cindex @code{asin()}
5858 @item @code{acos(x)}
5860 @cindex @code{acos()}
5861 @item @code{atan(x)}
5862 @tab inverse tangent
5863 @cindex @code{atan()}
5864 @item @code{atan2(y, x)}
5865 @tab inverse tangent with two arguments
5866 @item @code{sinh(x)}
5867 @tab hyperbolic sine
5868 @cindex @code{sinh()}
5869 @item @code{cosh(x)}
5870 @tab hyperbolic cosine
5871 @cindex @code{cosh()}
5872 @item @code{tanh(x)}
5873 @tab hyperbolic tangent
5874 @cindex @code{tanh()}
5875 @item @code{asinh(x)}
5876 @tab inverse hyperbolic sine
5877 @cindex @code{asinh()}
5878 @item @code{acosh(x)}
5879 @tab inverse hyperbolic cosine
5880 @cindex @code{acosh()}
5881 @item @code{atanh(x)}
5882 @tab inverse hyperbolic tangent
5883 @cindex @code{atanh()}
5885 @tab exponential function
5886 @cindex @code{exp()}
5888 @tab natural logarithm
5889 @cindex @code{log()}
5890 @item @code{eta(x,y)}
5891 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5892 @cindex @code{eta()}
5895 @cindex @code{Li2()}
5896 @item @code{Li(m, x)}
5897 @tab classical polylogarithm as well as multiple polylogarithm
5899 @item @code{G(a, y)}
5900 @tab multiple polylogarithm
5902 @item @code{G(a, s, y)}
5903 @tab multiple polylogarithm with explicit signs for the imaginary parts
5905 @item @code{S(n, p, x)}
5906 @tab Nielsen's generalized polylogarithm
5908 @item @code{H(m, x)}
5909 @tab harmonic polylogarithm
5911 @item @code{zeta(m)}
5912 @tab Riemann's zeta function as well as multiple zeta value
5913 @cindex @code{zeta()}
5914 @item @code{zeta(m, s)}
5915 @tab alternating Euler sum
5916 @cindex @code{zeta()}
5917 @item @code{zetaderiv(n, x)}
5918 @tab derivatives of Riemann's zeta function
5919 @item @code{iterated_integral(a, y)}
5920 @tab iterated integral
5921 @cindex @code{iterated_integral()}
5922 @item @code{iterated_integral(a, y, N)}
5923 @tab iterated integral with explicit truncation parameter
5924 @cindex @code{iterated_integral()}
5925 @item @code{tgamma(x)}
5927 @cindex @code{tgamma()}
5928 @cindex gamma function
5929 @item @code{lgamma(x)}
5930 @tab logarithm of gamma function
5931 @cindex @code{lgamma()}
5932 @item @code{beta(x, y)}
5933 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5934 @cindex @code{beta()}
5936 @tab psi (digamma) function
5937 @cindex @code{psi()}
5938 @item @code{psi(n, x)}
5939 @tab derivatives of psi function (polygamma functions)
5940 @item @code{EllipticK(x)}
5941 @tab complete elliptic integral of the first kind
5942 @cindex @code{EllipticK()}
5943 @item @code{EllipticE(x)}
5944 @tab complete elliptic integral of the second kind
5945 @cindex @code{EllipticE()}
5946 @item @code{factorial(n)}
5947 @tab factorial function @math{n!}
5948 @cindex @code{factorial()}
5949 @item @code{binomial(n, k)}
5950 @tab binomial coefficients
5951 @cindex @code{binomial()}
5952 @item @code{Order(x)}
5953 @tab order term function in truncated power series
5954 @cindex @code{Order()}
5959 For functions that have a branch cut in the complex plane, GiNaC
5960 follows the conventions of C/C++ for systems that do not support a
5961 signed zero. In particular: the natural logarithm (@code{log}) and
5962 the square root (@code{sqrt}) both have their branch cuts running
5963 along the negative real axis. The @code{asin}, @code{acos}, and
5964 @code{atanh} functions all have two branch cuts starting at +/-1 and
5965 running away towards infinity along the real axis. The @code{atan} and
5966 @code{asinh} functions have two branch cuts starting at +/-i and
5967 running away towards infinity along the imaginary axis. The
5968 @code{acosh} function has one branch cut starting at +1 and running
5969 towards -infinity. These functions are continuous as the branch cut
5970 is approached coming around the finite endpoint of the cut in a
5971 counter clockwise direction.
5974 @subsection Expanding functions
5975 @cindex expand trancedent functions
5976 @cindex @code{expand_options::expand_transcendental}
5977 @cindex @code{expand_options::expand_function_args}
5978 GiNaC knows several expansion laws for trancedent functions, e.g.
5984 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5988 $\log(c*d)=\log(c)+\log(d)$,
5991 @command{log(cd)=log(c)+log(d)}
6000 ). In order to use these rules you need to call @code{expand()} method
6001 with the option @code{expand_options::expand_transcendental}. Another
6002 relevant option is @code{expand_options::expand_function_args}. Their
6003 usage and interaction can be seen from the following example:
6006 symbol x("x"), y("y");
6007 ex e=exp(pow(x+y,2));
6008 cout << e.expand() << endl;
6010 cout << e.expand(expand_options::expand_transcendental) << endl;
6012 cout << e.expand(expand_options::expand_function_args) << endl;
6013 // -> exp(2*x*y+x^2+y^2)
6014 cout << e.expand(expand_options::expand_function_args
6015 | expand_options::expand_transcendental) << endl;
6016 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
6019 If both flags are set (as in the last call), then GiNaC tries to get
6020 the maximal expansion. For example, for the exponent GiNaC firstly expands
6021 the argument and then the function. For the logarithm and absolute value,
6022 GiNaC uses the opposite order: firstly expands the function and then its
6023 argument. Of course, a user can fine-tune this behavior by sequential
6024 calls of several @code{expand()} methods with desired flags.
6026 @node Multiple polylogarithms, Iterated integrals, Built-in functions, Methods and functions
6027 @c node-name, next, previous, up
6028 @subsection Multiple polylogarithms
6030 @cindex polylogarithm
6031 @cindex Nielsen's generalized polylogarithm
6032 @cindex harmonic polylogarithm
6033 @cindex multiple zeta value
6034 @cindex alternating Euler sum
6035 @cindex multiple polylogarithm
6037 The multiple polylogarithm is the most generic member of a family of functions,
6038 to which others like the harmonic polylogarithm, Nielsen's generalized
6039 polylogarithm and the multiple zeta value belong.
6040 Each of these functions can also be written as a multiple polylogarithm with specific
6041 parameters. This whole family of functions is therefore often referred to simply as
6042 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
6043 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
6044 @code{Li} and @code{G} in principle represent the same function, the different
6045 notations are more natural to the series representation or the integral
6046 representation, respectively.
6048 To facilitate the discussion of these functions we distinguish between indices and
6049 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
6050 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
6052 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
6053 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
6054 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6055 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6056 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6057 @code{s} is not given, the signs default to +1.
6058 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6059 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6060 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6061 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6062 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6064 The functions print in LaTeX format as
6066 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6072 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6075 $\zeta(m_1,m_2,\ldots,m_k)$.
6078 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6079 @command{\mbox@{S@}_@{n,p@}(x)},
6080 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6081 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6083 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6084 are printed with a line above, e.g.
6086 $\zeta(5,\overline{2})$.
6089 @command{\zeta(5,\overline@{2@})}.
6091 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6093 Definitions and analytical as well as numerical properties of multiple polylogarithms
6094 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6095 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6096 except for a few differences which will be explicitly stated in the following.
6098 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6099 that the indices and arguments are understood to be in the same order as in which they appear in
6100 the series representation. This means
6102 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6105 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6108 $\zeta(1,2)$ evaluates to infinity.
6111 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6112 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6113 @code{zeta(1,2)} evaluates to infinity.
6115 So in comparison to the older ones of the referenced publications the order of
6116 indices and arguments for @code{Li} is reversed.
6118 The functions only evaluate if the indices are integers greater than zero, except for the indices
6119 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6120 will be interpreted as the sequence of signs for the corresponding indices
6121 @code{m} or the sign of the imaginary part for the
6122 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6123 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6125 $\zeta(\overline{3},4)$
6128 @command{zeta(\overline@{3@},4)}
6131 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6133 $G(a-0\epsilon,b+0\epsilon;c)$.
6136 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6138 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6139 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6140 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
6141 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6142 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6143 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6144 evaluates also for negative integers and positive even integers. For example:
6147 > Li(@{3,1@},@{x,1@});
6150 -zeta(@{3,2@},@{-1,-1@})
6155 It is easy to tell for a given function into which other function it can be rewritten, may
6156 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6157 with negative indices or trailing zeros (the example above gives a hint). Signs can
6158 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6159 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6160 @code{Li} (@code{eval()} already cares for the possible downgrade):
6163 > convert_H_to_Li(@{0,-2,-1,3@},x);
6164 Li(@{3,1,3@},@{-x,1,-1@})
6165 > convert_H_to_Li(@{2,-1,0@},x);
6166 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6169 Every function can be numerically evaluated for
6170 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6171 global variable @code{Digits}:
6176 > evalf(zeta(@{3,1,3,1@}));
6177 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6180 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6181 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6183 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6191 In long expressions this helps a lot with debugging, because you can easily spot
6192 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6193 cancellations of divergencies happen.
6195 Useful publications:
6197 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6198 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6200 @cite{Harmonic Polylogarithms},
6201 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6203 @cite{Special Values of Multiple Polylogarithms},
6204 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6206 @cite{Numerical Evaluation of Multiple Polylogarithms},
6207 J.Vollinga, S.Weinzierl, hep-ph/0410259
6209 @node Iterated integrals, Complex expressions, Multiple polylogarithms, Methods and functions
6210 @c node-name, next, previous, up
6211 @subsection Iterated integrals
6213 Multiple polylogarithms are a particular example of iterated integrals.
6214 An iterated integral is defined by the function @code{iterated_integral(a,y)}.
6215 The variable @code{y} gives the upper integration limit for the outermost integration, by convention the lower integration limit is always set to zero.
6216 The variable @code{a} must be a GiNaC @code{lst} containing sub-classes of @code{integration_kernel} as elements.
6217 The depth of the iterated integral corresponds to the number of elements of @code{a}.
6218 The available integrands for iterated integrals are
6219 (for a more detailed description the user is referred to the publications listed at the end of this section)
6221 @multitable @columnfractions .40 .60
6222 @item @strong{Class} @tab @strong{Description}
6223 @item @code{integration_kernel()}
6224 @tab Base class, represents the one-form @math{dy}
6225 @cindex @code{integration_kernel()}
6226 @item @code{basic_log_kernel()}
6227 @tab Logarithmic one-form @math{dy/y}
6228 @cindex @code{basic_log_kernel()}
6229 @item @code{multiple_polylog_kernel(z_j)}
6230 @tab The one-form @math{dy/(y-z_j)}
6231 @cindex @code{multiple_polylog_kernel()}
6232 @item @code{ELi_kernel(n, m, x, y)}
6233 @tab The one form @math{ELi_{n;m}(x;y;q) dq/q}
6234 @cindex @code{ELi_kernel()}
6235 @item @code{Ebar_kernel(n, m, x, y)}
6236 @tab The one form @math{\overline{E}_{n;m}(x;y;q) dq/q}
6237 @cindex @code{Ebar_kernel()}
6238 @item @code{Kronecker_dtau_kernel(k, z_j, K, C_k)}
6239 @tab The one form @math{C_k K (k-1)/(2 \pi i)^k g^{(k)}(z_j,K \tau) dq/q}
6240 @cindex @code{Kronecker_dtau_kernel()}
6241 @item @code{Kronecker_dz_kernel(k, z_j, tau, K, C_k)}
6242 @tab The one form @math{C_k (2 \pi i)^{2-k} g^{(k-1)}(z-z_j,K \tau) dz}
6243 @cindex @code{Kronecker_dz_kernel()}
6244 @item @code{Eisenstein_kernel(k, N, a, b, K, C_k)}
6245 @tab The one form @math{C_k E_{k,N,a,b,K}(\tau) dq/q}
6246 @cindex @code{Eisenstein_kernel()}
6247 @item @code{Eisenstein_h_kernel(k, N, r, s, C_k)}
6248 @tab The one form @math{C_k h_{k,N,r,s}(\tau) dq/q}
6249 @cindex @code{Eisenstein_h_kernel()}
6250 @item @code{modular_form_kernel(k, P, C_k)}
6251 @tab The one form @math{C_k P dq/q}
6252 @cindex @code{modular_form_kernel()}
6253 @item @code{user_defined_kernel(f, y)}
6254 @tab The one form @math{f(y) dy}
6255 @cindex @code{user_defined_kernel()}
6258 All parameters are assumed to be such that all integration kernels have a convergent Laurent expansion
6259 around zero with at most a simple pole at zero.
6260 The iterated integral may also be called with an optional third parameter
6261 @code{iterated_integral(a,y,N_trunc)}, in which case the numerical evaluation will truncate the series
6262 expansion at order @code{N_trunc}.
6264 The classes @code{Eisenstein_kernel()}, @code{Eisenstein_h_kernel()} and @code{modular_form_kernel()}
6265 provide a method @code{q_expansion_modular_form(q, order)}, which can used to obtain the q-expansion
6266 of @math{E_{k,N,a,b,K}(\tau)}, @math{h_{k,N,r,s}(\tau)} or @math{P} to the specified order.
6268 Useful publications:
6270 @cite{Numerical evaluation of iterated integrals related to elliptic Feynman integrals},
6271 M.Walden, S.Weinzierl, arXiv:2010.05271
6273 @node Complex expressions, Solving linear systems of equations, Iterated integrals, Methods and functions
6274 @c node-name, next, previous, up
6275 @section Complex expressions
6277 @cindex @code{conjugate()}
6279 For dealing with complex expressions there are the methods
6287 that return respectively the complex conjugate, the real part and the
6288 imaginary part of an expression. Complex conjugation works as expected
6289 for all built-in functions and objects. Taking real and imaginary
6290 parts has not yet been implemented for all built-in functions. In cases where
6291 it is not known how to conjugate or take a real/imaginary part one
6292 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6293 is returned. For instance, in case of a complex symbol @code{x}
6294 (symbols are complex by default), one could not simplify
6295 @code{conjugate(x)}. In the case of strings of gamma matrices,
6296 the @code{conjugate} method takes the Dirac conjugate.
6301 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6305 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6306 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6307 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6308 // -> -gamma5*gamma~b*gamma~a
6312 If you declare your own GiNaC functions and you want to conjugate them, you
6313 will have to supply a specialized conjugation method for them (see
6314 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6315 example). GiNaC does not automatically conjugate user-supplied functions
6316 by conjugating their arguments because this would be incorrect on branch
6317 cuts. Also, specialized methods can be provided to take real and imaginary
6318 parts of user-defined functions.
6320 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6321 @c node-name, next, previous, up
6322 @section Solving linear systems of equations
6323 @cindex @code{lsolve()}
6325 The function @code{lsolve()} provides a convenient wrapper around some
6326 matrix operations that comes in handy when a system of linear equations
6330 ex lsolve(const ex & eqns, const ex & symbols,
6331 unsigned options = solve_algo::automatic);
6334 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6335 @code{relational}) while @code{symbols} is a @code{lst} of
6336 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6339 It returns the @code{lst} of solutions as an expression. As an example,
6340 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6344 symbol a("a"), b("b"), x("x"), y("y");
6345 lst eqns = @{a*x+b*y==3, x-y==b@};
6346 lst vars = @{x, y@};
6347 cout << lsolve(eqns, vars) << endl;
6348 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6351 When the linear equations @code{eqns} are underdetermined, the solution
6352 will contain one or more tautological entries like @code{x==x},
6353 depending on the rank of the system. When they are overdetermined, the
6354 solution will be an empty @code{lst}. Note the third optional parameter
6355 to @code{lsolve()}: it accepts the same parameters as
6356 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6360 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6361 @c node-name, next, previous, up
6362 @section Input and output of expressions
6365 @subsection Expression output
6367 @cindex output of expressions
6369 Expressions can simply be written to any stream:
6374 ex e = 4.5*I+pow(x,2)*3/2;
6375 cout << e << endl; // prints '4.5*I+3/2*x^2'
6379 The default output format is identical to the @command{ginsh} input syntax and
6380 to that used by most computer algebra systems, but not directly pastable
6381 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6382 is printed as @samp{x^2}).
6384 It is possible to print expressions in a number of different formats with
6385 a set of stream manipulators;
6388 std::ostream & dflt(std::ostream & os);
6389 std::ostream & latex(std::ostream & os);
6390 std::ostream & tree(std::ostream & os);
6391 std::ostream & csrc(std::ostream & os);
6392 std::ostream & csrc_float(std::ostream & os);
6393 std::ostream & csrc_double(std::ostream & os);
6394 std::ostream & csrc_cl_N(std::ostream & os);
6395 std::ostream & index_dimensions(std::ostream & os);
6396 std::ostream & no_index_dimensions(std::ostream & os);
6399 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6400 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6401 @code{print_csrc()} functions, respectively.
6404 All manipulators affect the stream state permanently. To reset the output
6405 format to the default, use the @code{dflt} manipulator:
6409 cout << latex; // all output to cout will be in LaTeX format from
6411 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6412 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6413 cout << dflt; // revert to default output format
6414 cout << e << endl; // prints '4.5*I+3/2*x^2'
6418 If you don't want to affect the format of the stream you're working with,
6419 you can output to a temporary @code{ostringstream} like this:
6424 s << latex << e; // format of cout remains unchanged
6425 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6429 @anchor{csrc printing}
6431 @cindex @code{csrc_float}
6432 @cindex @code{csrc_double}
6433 @cindex @code{csrc_cl_N}
6434 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6435 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6436 format that can be directly used in a C or C++ program. The three possible
6437 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6438 classes provided by the CLN library):
6442 cout << "f = " << csrc_float << e << ";\n";
6443 cout << "d = " << csrc_double << e << ";\n";
6444 cout << "n = " << csrc_cl_N << e << ";\n";
6448 The above example will produce (note the @code{x^2} being converted to
6452 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6453 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6454 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6458 The @code{tree} manipulator allows dumping the internal structure of an
6459 expression for debugging purposes:
6470 add, hash=0x0, flags=0x3, nops=2
6471 power, hash=0x0, flags=0x3, nops=2
6472 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6473 2 (numeric), hash=0x6526b0fa, flags=0xf
6474 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6477 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6481 @cindex @code{latex}
6482 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6483 It is rather similar to the default format but provides some braces needed
6484 by LaTeX for delimiting boxes and also converts some common objects to
6485 conventional LaTeX names. It is possible to give symbols a special name for
6486 LaTeX output by supplying it as a second argument to the @code{symbol}
6489 For example, the code snippet
6493 symbol x("x", "\\circ");
6494 ex e = lgamma(x).series(x==0,3);
6495 cout << latex << e << endl;
6502 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6503 +\mathcal@{O@}(\circ^@{3@})
6506 @cindex @code{index_dimensions}
6507 @cindex @code{no_index_dimensions}
6508 Index dimensions are normally hidden in the output. To make them visible, use
6509 the @code{index_dimensions} manipulator. The dimensions will be written in
6510 square brackets behind each index value in the default and LaTeX output
6515 symbol x("x"), y("y");
6516 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6517 ex e = indexed(x, mu) * indexed(y, nu);
6520 // prints 'x~mu*y~nu'
6521 cout << index_dimensions << e << endl;
6522 // prints 'x~mu[4]*y~nu[4]'
6523 cout << no_index_dimensions << e << endl;
6524 // prints 'x~mu*y~nu'
6529 @cindex Tree traversal
6530 If you need any fancy special output format, e.g. for interfacing GiNaC
6531 with other algebra systems or for producing code for different
6532 programming languages, you can always traverse the expression tree yourself:
6535 static void my_print(const ex & e)
6537 if (is_a<function>(e))
6538 cout << ex_to<function>(e).get_name();
6540 cout << ex_to<basic>(e).class_name();
6542 size_t n = e.nops();
6544 for (size_t i=0; i<n; i++) @{
6556 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6564 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6565 symbol(y))),numeric(-2)))
6568 If you need an output format that makes it possible to accurately
6569 reconstruct an expression by feeding the output to a suitable parser or
6570 object factory, you should consider storing the expression in an
6571 @code{archive} object and reading the object properties from there.
6572 See the section on archiving for more information.
6575 @subsection Expression input
6576 @cindex input of expressions
6578 GiNaC provides no way to directly read an expression from a stream because
6579 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6580 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6581 @code{y} you defined in your program and there is no way to specify the
6582 desired symbols to the @code{>>} stream input operator.
6584 Instead, GiNaC lets you read an expression from a stream or a string,
6585 specifying the mapping between the input strings and symbols to be used:
6593 parser reader(table);
6594 ex e = reader("2*x+sin(y)");
6598 The input syntax is the same as that used by @command{ginsh} and the stream
6599 output operator @code{<<}. Matching between the input strings and expressions
6600 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6601 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6602 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6603 to map input (sub)strings to arbitrary expressions:
6609 table["x"] = x+log(y)+1;
6610 parser reader(table);
6611 ex e = reader("5*x^3 - x^2");
6612 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6616 If no mapping is specified for a particular string GiNaC will create a symbol
6617 with corresponding name. Later on you can obtain all parser generated symbols
6618 with @code{get_syms()} method:
6623 ex e = reader("2*x+sin(y)");
6624 symtab table = reader.get_syms();
6625 symbol x = ex_to<symbol>(table["x"]);
6626 symbol y = ex_to<symbol>(table["y"]);
6630 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6631 (for example, you want treat an unexpected string in the input as an error).
6636 table["x"] = symbol();
6637 parser reader(table);
6638 parser.strict = true;
6641 e = reader("2*x+sin(y)");
6642 @} catch (parse_error& err) @{
6643 cerr << err.what() << endl;
6644 // prints "unknown symbol "y" in the input"
6649 With this parser, it's also easy to implement interactive GiNaC programs.
6650 When running the following program interactively, remember to send an
6651 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6656 #include <stdexcept>
6657 #include <ginac/ginac.h>
6658 using namespace std;
6659 using namespace GiNaC;
6663 cout << "Enter an expression containing 'x': " << flush;
6668 symtab table = reader.get_syms();
6669 symbol x = table.find("x") != table.end() ?
6670 ex_to<symbol>(table["x"]) : symbol("x");
6671 cout << "The derivative of " << e << " with respect to x is ";
6672 cout << e.diff(x) << "." << endl;
6673 @} catch (exception &p) @{
6674 cerr << p.what() << endl;
6679 @subsection Compiling expressions to C function pointers
6680 @cindex compiling expressions
6682 Numerical evaluation of algebraic expressions is seamlessly integrated into
6683 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6684 precision numerics, which is more than sufficient for most users, sometimes only
6685 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6686 Carlo integration. The only viable option then is the following: print the
6687 expression in C syntax format, manually add necessary C code, compile that
6688 program and run is as a separate application. This is not only cumbersome and
6689 involves a lot of manual intervention, but it also separates the algebraic and
6690 the numerical evaluation into different execution stages.
6692 GiNaC offers a couple of functions that help to avoid these inconveniences and
6693 problems. The functions automatically perform the printing of a GiNaC expression
6694 and the subsequent compiling of its associated C code. The created object code
6695 is then dynamically linked to the currently running program. A function pointer
6696 to the C function that performs the numerical evaluation is returned and can be
6697 used instantly. This all happens automatically, no user intervention is needed.
6699 The following example demonstrates the use of @code{compile_ex}:
6704 ex myexpr = sin(x) / x;
6707 compile_ex(myexpr, x, fp);
6709 cout << fp(3.2) << endl;
6713 The function @code{compile_ex} is called with the expression to be compiled and
6714 its only free variable @code{x}. Upon successful completion the third parameter
6715 contains a valid function pointer to the corresponding C code module. If called
6716 like in the last line only built-in double precision numerics is involved.
6721 The function pointer has to be defined in advance. GiNaC offers three function
6722 pointer types at the moment:
6725 typedef double (*FUNCP_1P) (double);
6726 typedef double (*FUNCP_2P) (double, double);
6727 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6730 @cindex CUBA library
6731 @cindex Monte Carlo integration
6732 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6733 the correct type to be used with the CUBA library
6734 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6735 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6738 For every function pointer type there is a matching @code{compile_ex} available:
6741 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6742 const std::string filename = "");
6743 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6744 FUNCP_2P& fp, const std::string filename = "");
6745 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6746 const std::string filename = "");
6749 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6750 choose a unique random name for the intermediate source and object files it
6751 produces. On program termination these files will be deleted. If one wishes to
6752 keep the C code and the object files, one can supply the @code{filename}
6753 parameter. The intermediate files will use that filename and will not be
6757 @code{link_ex} is a function that allows to dynamically link an existing object
6758 file and to make it available via a function pointer. This is useful if you
6759 have already used @code{compile_ex} on an expression and want to avoid the
6760 compilation step to be performed over and over again when you restart your
6761 program. The precondition for this is of course, that you have chosen a
6762 filename when you did call @code{compile_ex}. For every above mentioned
6763 function pointer type there exists a corresponding @code{link_ex} function:
6766 void link_ex(const std::string filename, FUNCP_1P& fp);
6767 void link_ex(const std::string filename, FUNCP_2P& fp);
6768 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6771 The complete filename (including the suffix @code{.so}) of the object file has
6778 void unlink_ex(const std::string filename);
6781 is supplied for the rare cases when one wishes to close the dynamically linked
6782 object files directly and have the intermediate files (only if filename has not
6783 been given) deleted. Normally one doesn't need this function, because all the
6784 clean-up will be done automatically upon (regular) program termination.
6786 All the described functions will throw an exception in case they cannot perform
6787 correctly, like for example when writing the file or starting the compiler
6788 fails. Since internally the same printing methods as described in section
6789 @ref{csrc printing} are used, only functions and objects that are available in
6790 standard C will compile successfully (that excludes polylogarithms for example
6791 at the moment). Another precondition for success is, of course, that it must be
6792 possible to evaluate the expression numerically. No free variables despite the
6793 ones supplied to @code{compile_ex} should appear in the expression.
6795 @cindex ginac-excompiler
6796 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6797 compiler and produce the object files. This shell script comes with GiNaC and
6798 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6799 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6800 export additional compiler flags via the @env{$CXXFLAGS} variable:
6803 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6807 @subsection Archiving
6808 @cindex @code{archive} (class)
6811 GiNaC allows creating @dfn{archives} of expressions which can be stored
6812 to or retrieved from files. To create an archive, you declare an object
6813 of class @code{archive} and archive expressions in it, giving each
6814 expression a unique name:
6818 #include <ginac/ginac.h>
6819 using namespace std;
6820 using namespace GiNaC;
6824 symbol x("x"), y("y"), z("z");
6826 ex foo = sin(x + 2*y) + 3*z + 41;
6830 a.archive_ex(foo, "foo");
6831 a.archive_ex(bar, "the second one");
6835 The archive can then be written to a file:
6839 ofstream out("foobar.gar", ios::binary);
6845 The file @file{foobar.gar} contains all information that is needed to
6846 reconstruct the expressions @code{foo} and @code{bar}. The flag
6847 @code{ios::binary} prevents locales setting of your OS tampers the
6848 archive file structure.
6850 @cindex @command{viewgar}
6851 The tool @command{viewgar} that comes with GiNaC can be used to view
6852 the contents of GiNaC archive files:
6855 $ viewgar foobar.gar
6856 foo = 41+sin(x+2*y)+3*z
6857 the second one = 42+sin(x+2*y)+3*z
6860 The point of writing archive files is of course that they can later be
6866 ifstream in("foobar.gar", ios::binary);
6871 And the stored expressions can be retrieved by their name:
6875 lst syms = @{x, y@};
6877 ex ex1 = a2.unarchive_ex(syms, "foo");
6878 ex ex2 = a2.unarchive_ex(syms, "the second one");
6880 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6881 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6882 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6886 Note that you have to supply a list of the symbols which are to be inserted
6887 in the expressions. Symbols in archives are stored by their name only and
6888 if you don't specify which symbols you have, unarchiving the expression will
6889 create new symbols with that name. E.g. if you hadn't included @code{x} in
6890 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6891 have had no effect because the @code{x} in @code{ex1} would have been a
6892 different symbol than the @code{x} which was defined at the beginning of
6893 the program, although both would appear as @samp{x} when printed.
6895 You can also use the information stored in an @code{archive} object to
6896 output expressions in a format suitable for exact reconstruction. The
6897 @code{archive} and @code{archive_node} classes have a couple of member
6898 functions that let you access the stored properties:
6901 static void my_print2(const archive_node & n)
6904 n.find_string("class", class_name);
6905 cout << class_name << "(";
6907 archive_node::propinfovector p;
6908 n.get_properties(p);
6910 size_t num = p.size();
6911 for (size_t i=0; i<num; i++) @{
6912 const string &name = p[i].name;
6913 if (name == "class")
6915 cout << name << "=";
6917 unsigned count = p[i].count;
6921 for (unsigned j=0; j<count; j++) @{
6922 switch (p[i].type) @{
6923 case archive_node::PTYPE_BOOL: @{
6925 n.find_bool(name, x, j);
6926 cout << (x ? "true" : "false");
6929 case archive_node::PTYPE_UNSIGNED: @{
6931 n.find_unsigned(name, x, j);
6935 case archive_node::PTYPE_STRING: @{
6937 n.find_string(name, x, j);
6938 cout << '\"' << x << '\"';
6941 case archive_node::PTYPE_NODE: @{
6942 const archive_node &x = n.find_ex_node(name, j);
6964 ex e = pow(2, x) - y;
6966 my_print2(ar.get_top_node(0)); cout << endl;
6974 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6975 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6976 overall_coeff=numeric(number="0"))
6979 Be warned, however, that the set of properties and their meaning for each
6980 class may change between GiNaC versions.
6983 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6984 @c node-name, next, previous, up
6985 @chapter Extending GiNaC
6987 By reading so far you should have gotten a fairly good understanding of
6988 GiNaC's design patterns. From here on you should start reading the
6989 sources. All we can do now is issue some recommendations how to tackle
6990 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6991 develop some useful extension please don't hesitate to contact the GiNaC
6992 authors---they will happily incorporate them into future versions.
6995 * What does not belong into GiNaC:: What to avoid.
6996 * Symbolic functions:: Implementing symbolic functions.
6997 * Printing:: Adding new output formats.
6998 * Structures:: Defining new algebraic classes (the easy way).
6999 * Adding classes:: Defining new algebraic classes (the hard way).
7003 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
7004 @c node-name, next, previous, up
7005 @section What doesn't belong into GiNaC
7007 @cindex @command{ginsh}
7008 First of all, GiNaC's name must be read literally. It is designed to be
7009 a library for use within C++. The tiny @command{ginsh} accompanying
7010 GiNaC makes this even more clear: it doesn't even attempt to provide a
7011 language. There are no loops or conditional expressions in
7012 @command{ginsh}, it is merely a window into the library for the
7013 programmer to test stuff (or to show off). Still, the design of a
7014 complete CAS with a language of its own, graphical capabilities and all
7015 this on top of GiNaC is possible and is without doubt a nice project for
7018 There are many built-in functions in GiNaC that do not know how to
7019 evaluate themselves numerically to a precision declared at runtime
7020 (using @code{Digits}). Some may be evaluated at certain points, but not
7021 generally. This ought to be fixed. However, doing numerical
7022 computations with GiNaC's quite abstract classes is doomed to be
7023 inefficient. For this purpose, the underlying foundation classes
7024 provided by CLN are much better suited.
7027 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
7028 @c node-name, next, previous, up
7029 @section Symbolic functions
7031 The easiest and most instructive way to start extending GiNaC is probably to
7032 create your own symbolic functions. These are implemented with the help of
7033 two preprocessor macros:
7035 @cindex @code{DECLARE_FUNCTION}
7036 @cindex @code{REGISTER_FUNCTION}
7038 DECLARE_FUNCTION_<n>P(<name>)
7039 REGISTER_FUNCTION(<name>, <options>)
7042 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
7043 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
7044 parameters of type @code{ex} and returns a newly constructed GiNaC
7045 @code{function} object that represents your function.
7047 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
7048 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
7049 set of options that associate the symbolic function with C++ functions you
7050 provide to implement the various methods such as evaluation, derivative,
7051 series expansion etc. They also describe additional attributes the function
7052 might have, such as symmetry and commutation properties, and a name for
7053 LaTeX output. Multiple options are separated by the member access operator
7054 @samp{.} and can be given in an arbitrary order.
7056 (By the way: in case you are worrying about all the macros above we can
7057 assure you that functions are GiNaC's most macro-intense classes. We have
7058 done our best to avoid macros where we can.)
7060 @subsection A minimal example
7062 Here is an example for the implementation of a function with two arguments
7063 that is not further evaluated:
7066 DECLARE_FUNCTION_2P(myfcn)
7068 REGISTER_FUNCTION(myfcn, dummy())
7071 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
7072 in algebraic expressions:
7078 ex e = 2*myfcn(42, 1+3*x) - x;
7080 // prints '2*myfcn(42,1+3*x)-x'
7085 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
7086 "no options". A function with no options specified merely acts as a kind of
7087 container for its arguments. It is a pure "dummy" function with no associated
7088 logic (which is, however, sometimes perfectly sufficient).
7090 Let's now have a look at the implementation of GiNaC's cosine function for an
7091 example of how to make an "intelligent" function.
7093 @subsection The cosine function
7095 The GiNaC header file @file{inifcns.h} contains the line
7098 DECLARE_FUNCTION_1P(cos)
7101 which declares to all programs using GiNaC that there is a function @samp{cos}
7102 that takes one @code{ex} as an argument. This is all they need to know to use
7103 this function in expressions.
7105 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
7106 is its @code{REGISTER_FUNCTION} line:
7109 REGISTER_FUNCTION(cos, eval_func(cos_eval).
7110 evalf_func(cos_evalf).
7111 derivative_func(cos_deriv).
7112 latex_name("\\cos"));
7115 There are four options defined for the cosine function. One of them
7116 (@code{latex_name}) gives the function a proper name for LaTeX output; the
7117 other three indicate the C++ functions in which the "brains" of the cosine
7118 function are defined.
7120 @cindex @code{hold()}
7122 The @code{eval_func()} option specifies the C++ function that implements
7123 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7124 the same number of arguments as the associated symbolic function (one in this
7125 case) and returns the (possibly transformed or in some way simplified)
7126 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7127 of the automatic evaluation process). If no (further) evaluation is to take
7128 place, the @code{eval_func()} function must return the original function
7129 with @code{.hold()}, to avoid a potential infinite recursion. If your
7130 symbolic functions produce a segmentation fault or stack overflow when
7131 using them in expressions, you are probably missing a @code{.hold()}
7134 The @code{eval_func()} function for the cosine looks something like this
7135 (actually, it doesn't look like this at all, but it should give you an idea
7139 static ex cos_eval(const ex & x)
7141 if ("x is a multiple of 2*Pi")
7143 else if ("x is a multiple of Pi")
7145 else if ("x is a multiple of Pi/2")
7149 else if ("x has the form 'acos(y)'")
7151 else if ("x has the form 'asin(y)'")
7156 return cos(x).hold();
7160 This function is called every time the cosine is used in a symbolic expression:
7166 // this calls cos_eval(Pi), and inserts its return value into
7167 // the actual expression
7174 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7175 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7176 symbolic transformation can be done, the unmodified function is returned
7177 with @code{.hold()}.
7179 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7180 The user has to call @code{evalf()} for that. This is implemented in a
7184 static ex cos_evalf(const ex & x)
7186 if (is_a<numeric>(x))
7187 return cos(ex_to<numeric>(x));
7189 return cos(x).hold();
7193 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7194 in this case the @code{cos()} function for @code{numeric} objects, which in
7195 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7196 isn't really needed here, but reminds us that the corresponding @code{eval()}
7197 function would require it in this place.
7199 Differentiation will surely turn up and so we need to tell @code{cos}
7200 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7201 instance, are then handled automatically by @code{basic::diff} and
7205 static ex cos_deriv(const ex & x, unsigned diff_param)
7211 @cindex product rule
7212 The second parameter is obligatory but uninteresting at this point. It
7213 specifies which parameter to differentiate in a partial derivative in
7214 case the function has more than one parameter, and its main application
7215 is for correct handling of the chain rule.
7217 Derivatives of some functions, for example @code{abs()} and
7218 @code{Order()}, could not be evaluated through the chain rule. In such
7219 cases the full derivative may be specified as shown for @code{Order()}:
7222 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7224 return Order(arg.diff(s));
7228 That is, we need to supply a procedure, which returns the expression of
7229 derivative with respect to the variable @code{s} for the argument
7230 @code{arg}. This procedure need to be registered with the function
7231 through the option @code{expl_derivative_func} (see the next
7232 Subsection). In contrast, a partial derivative, e.g. as was defined for
7233 @code{cos()} above, needs to be registered through the option
7234 @code{derivative_func}.
7236 An implementation of the series expansion is not needed for @code{cos()} as
7237 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7238 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7239 the other hand, does have poles and may need to do Laurent expansion:
7242 static ex tan_series(const ex & x, const relational & rel,
7243 int order, unsigned options)
7245 // Find the actual expansion point
7246 const ex x_pt = x.subs(rel);
7248 if ("x_pt is not an odd multiple of Pi/2")
7249 throw do_taylor(); // tell function::series() to do Taylor expansion
7251 // On a pole, expand sin()/cos()
7252 return (sin(x)/cos(x)).series(rel, order+2, options);
7256 The @code{series()} implementation of a function @emph{must} return a
7257 @code{pseries} object, otherwise your code will crash.
7259 @subsection Function options
7261 GiNaC functions understand several more options which are always
7262 specified as @code{.option(params)}. None of them are required, but you
7263 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7264 is a do-nothing option called @code{dummy()} which you can use to define
7265 functions without any special options.
7268 eval_func(<C++ function>)
7269 evalf_func(<C++ function>)
7270 derivative_func(<C++ function>)
7271 expl_derivative_func(<C++ function>)
7272 series_func(<C++ function>)
7273 conjugate_func(<C++ function>)
7276 These specify the C++ functions that implement symbolic evaluation,
7277 numeric evaluation, partial derivatives, explicit derivative, and series
7278 expansion, respectively. They correspond to the GiNaC methods
7279 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7281 The @code{eval_func()} function needs to use @code{.hold()} if no further
7282 automatic evaluation is desired or possible.
7284 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7285 expansion, which is correct if there are no poles involved. If the function
7286 has poles in the complex plane, the @code{series_func()} needs to check
7287 whether the expansion point is on a pole and fall back to Taylor expansion
7288 if it isn't. Otherwise, the pole usually needs to be regularized by some
7289 suitable transformation.
7292 latex_name(const string & n)
7295 specifies the LaTeX code that represents the name of the function in LaTeX
7296 output. The default is to put the function name in an @code{\mbox@{@}}.
7299 do_not_evalf_params()
7302 This tells @code{evalf()} to not recursively evaluate the parameters of the
7303 function before calling the @code{evalf_func()}.
7306 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7309 This allows you to explicitly specify the commutation properties of the
7310 function (@xref{Non-commutative objects}, for an explanation of
7311 (non)commutativity in GiNaC). For example, with an object of type
7312 @code{return_type_t} created like
7315 return_type_t my_type = make_return_type_t<matrix>();
7318 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7319 make GiNaC treat your function like a matrix. By default, functions inherit the
7320 commutation properties of their first argument. The utilized template function
7321 @code{make_return_type_t<>()}
7324 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7327 can also be called with an argument specifying the representation label of the
7328 non-commutative function (see section on dirac gamma matrices for more
7332 set_symmetry(const symmetry & s)
7335 specifies the symmetry properties of the function with respect to its
7336 arguments. @xref{Indexed objects}, for an explanation of symmetry
7337 specifications. GiNaC will automatically rearrange the arguments of
7338 symmetric functions into a canonical order.
7340 Sometimes you may want to have finer control over how functions are
7341 displayed in the output. For example, the @code{abs()} function prints
7342 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7343 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7347 print_func<C>(<C++ function>)
7350 option which is explained in the next section.
7352 @subsection Functions with a variable number of arguments
7354 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7355 functions with a fixed number of arguments. Sometimes, though, you may need
7356 to have a function that accepts a variable number of expressions. One way to
7357 accomplish this is to pass variable-length lists as arguments. The
7358 @code{Li()} function uses this method for multiple polylogarithms.
7360 It is also possible to define functions that accept a different number of
7361 parameters under the same function name, such as the @code{psi()} function
7362 which can be called either as @code{psi(z)} (the digamma function) or as
7363 @code{psi(n, z)} (polygamma functions). These are actually two different
7364 functions in GiNaC that, however, have the same name. Defining such
7365 functions is not possible with the macros but requires manually fiddling
7366 with GiNaC internals. If you are interested, please consult the GiNaC source
7367 code for the @code{psi()} function (@file{inifcns.h} and
7368 @file{inifcns_gamma.cpp}).
7371 @node Printing, Structures, Symbolic functions, Extending GiNaC
7372 @c node-name, next, previous, up
7373 @section GiNaC's expression output system
7375 GiNaC allows the output of expressions in a variety of different formats
7376 (@pxref{Input/output}). This section will explain how expression output
7377 is implemented internally, and how to define your own output formats or
7378 change the output format of built-in algebraic objects. You will also want
7379 to read this section if you plan to write your own algebraic classes or
7382 @cindex @code{print_context} (class)
7383 @cindex @code{print_dflt} (class)
7384 @cindex @code{print_latex} (class)
7385 @cindex @code{print_tree} (class)
7386 @cindex @code{print_csrc} (class)
7387 All the different output formats are represented by a hierarchy of classes
7388 rooted in the @code{print_context} class, defined in the @file{print.h}
7393 the default output format
7395 output in LaTeX mathematical mode
7397 a dump of the internal expression structure (for debugging)
7399 the base class for C source output
7400 @item print_csrc_float
7401 C source output using the @code{float} type
7402 @item print_csrc_double
7403 C source output using the @code{double} type
7404 @item print_csrc_cl_N
7405 C source output using CLN types
7408 The @code{print_context} base class provides two public data members:
7420 @code{s} is a reference to the stream to output to, while @code{options}
7421 holds flags and modifiers. Currently, there is only one flag defined:
7422 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7423 to print the index dimension which is normally hidden.
7425 When you write something like @code{std::cout << e}, where @code{e} is
7426 an object of class @code{ex}, GiNaC will construct an appropriate
7427 @code{print_context} object (of a class depending on the selected output
7428 format), fill in the @code{s} and @code{options} members, and call
7430 @cindex @code{print()}
7432 void ex::print(const print_context & c, unsigned level = 0) const;
7435 which in turn forwards the call to the @code{print()} method of the
7436 top-level algebraic object contained in the expression.
7438 Unlike other methods, GiNaC classes don't usually override their
7439 @code{print()} method to implement expression output. Instead, the default
7440 implementation @code{basic::print(c, level)} performs a run-time double
7441 dispatch to a function selected by the dynamic type of the object and the
7442 passed @code{print_context}. To this end, GiNaC maintains a separate method
7443 table for each class, similar to the virtual function table used for ordinary
7444 (single) virtual function dispatch.
7446 The method table contains one slot for each possible @code{print_context}
7447 type, indexed by the (internally assigned) serial number of the type. Slots
7448 may be empty, in which case GiNaC will retry the method lookup with the
7449 @code{print_context} object's parent class, possibly repeating the process
7450 until it reaches the @code{print_context} base class. If there's still no
7451 method defined, the method table of the algebraic object's parent class
7452 is consulted, and so on, until a matching method is found (eventually it
7453 will reach the combination @code{basic/print_context}, which prints the
7454 object's class name enclosed in square brackets).
7456 You can think of the print methods of all the different classes and output
7457 formats as being arranged in a two-dimensional matrix with one axis listing
7458 the algebraic classes and the other axis listing the @code{print_context}
7461 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7462 to implement printing, but then they won't get any of the benefits of the
7463 double dispatch mechanism (such as the ability for derived classes to
7464 inherit only certain print methods from its parent, or the replacement of
7465 methods at run-time).
7467 @subsection Print methods for classes
7469 The method table for a class is set up either in the definition of the class,
7470 by passing the appropriate @code{print_func<C>()} option to
7471 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7472 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7473 can also be used to override existing methods dynamically.
7475 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7476 be a member function of the class (or one of its parent classes), a static
7477 member function, or an ordinary (global) C++ function. The @code{C} template
7478 parameter specifies the appropriate @code{print_context} type for which the
7479 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7480 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7481 the class is the one being implemented by
7482 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7484 For print methods that are member functions, their first argument must be of
7485 a type convertible to a @code{const C &}, and the second argument must be an
7488 For static members and global functions, the first argument must be of a type
7489 convertible to a @code{const T &}, the second argument must be of a type
7490 convertible to a @code{const C &}, and the third argument must be an
7491 @code{unsigned}. A global function will, of course, not have access to
7492 private and protected members of @code{T}.
7494 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7495 and @code{basic::print()}) is used for proper parenthesizing of the output
7496 (and by @code{print_tree} for proper indentation). It can be used for similar
7497 purposes if you write your own output formats.
7499 The explanations given above may seem complicated, but in practice it's
7500 really simple, as shown in the following example. Suppose that we want to
7501 display exponents in LaTeX output not as superscripts but with little
7502 upwards-pointing arrows. This can be achieved in the following way:
7505 void my_print_power_as_latex(const power & p,
7506 const print_latex & c,
7509 // get the precedence of the 'power' class
7510 unsigned power_prec = p.precedence();
7512 // if the parent operator has the same or a higher precedence
7513 // we need parentheses around the power
7514 if (level >= power_prec)
7517 // print the basis and exponent, each enclosed in braces, and
7518 // separated by an uparrow
7520 p.op(0).print(c, power_prec);
7521 c.s << "@}\\uparrow@{";
7522 p.op(1).print(c, power_prec);
7525 // don't forget the closing parenthesis
7526 if (level >= power_prec)
7532 // a sample expression
7533 symbol x("x"), y("y");
7534 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7536 // switch to LaTeX mode
7539 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7542 // now we replace the method for the LaTeX output of powers with
7544 set_print_func<power, print_latex>(my_print_power_as_latex);
7546 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7557 The first argument of @code{my_print_power_as_latex} could also have been
7558 a @code{const basic &}, the second one a @code{const print_context &}.
7561 The above code depends on @code{mul} objects converting their operands to
7562 @code{power} objects for the purpose of printing.
7565 The output of products including negative powers as fractions is also
7566 controlled by the @code{mul} class.
7569 The @code{power/print_latex} method provided by GiNaC prints square roots
7570 using @code{\sqrt}, but the above code doesn't.
7574 It's not possible to restore a method table entry to its previous or default
7575 value. Once you have called @code{set_print_func()}, you can only override
7576 it with another call to @code{set_print_func()}, but you can't easily go back
7577 to the default behavior again (you can, of course, dig around in the GiNaC
7578 sources, find the method that is installed at startup
7579 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7580 one; that is, after you circumvent the C++ member access control@dots{}).
7582 @subsection Print methods for functions
7584 Symbolic functions employ a print method dispatch mechanism similar to the
7585 one used for classes. The methods are specified with @code{print_func<C>()}
7586 function options. If you don't specify any special print methods, the function
7587 will be printed with its name (or LaTeX name, if supplied), followed by a
7588 comma-separated list of arguments enclosed in parentheses.
7590 For example, this is what GiNaC's @samp{abs()} function is defined like:
7593 static ex abs_eval(const ex & arg) @{ ... @}
7594 static ex abs_evalf(const ex & arg) @{ ... @}
7596 static void abs_print_latex(const ex & arg, const print_context & c)
7598 c.s << "@{|"; arg.print(c); c.s << "|@}";
7601 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7603 c.s << "fabs("; arg.print(c); c.s << ")";
7606 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7607 evalf_func(abs_evalf).
7608 print_func<print_latex>(abs_print_latex).
7609 print_func<print_csrc_float>(abs_print_csrc_float).
7610 print_func<print_csrc_double>(abs_print_csrc_float));
7613 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7614 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7616 There is currently no equivalent of @code{set_print_func()} for functions.
7618 @subsection Adding new output formats
7620 Creating a new output format involves subclassing @code{print_context},
7621 which is somewhat similar to adding a new algebraic class
7622 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7623 that needs to go into the class definition, and a corresponding macro
7624 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7625 Every @code{print_context} class needs to provide a default constructor
7626 and a constructor from an @code{std::ostream} and an @code{unsigned}
7629 Here is an example for a user-defined @code{print_context} class:
7632 class print_myformat : public print_dflt
7634 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7636 print_myformat(std::ostream & os, unsigned opt = 0)
7637 : print_dflt(os, opt) @{@}
7640 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7642 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7645 That's all there is to it. None of the actual expression output logic is
7646 implemented in this class. It merely serves as a selector for choosing
7647 a particular format. The algorithms for printing expressions in the new
7648 format are implemented as print methods, as described above.
7650 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7651 exactly like GiNaC's default output format:
7656 ex e = pow(x, 2) + 1;
7658 // this prints "1+x^2"
7661 // this also prints "1+x^2"
7662 e.print(print_myformat()); cout << endl;
7668 To fill @code{print_myformat} with life, we need to supply appropriate
7669 print methods with @code{set_print_func()}, like this:
7672 // This prints powers with '**' instead of '^'. See the LaTeX output
7673 // example above for explanations.
7674 void print_power_as_myformat(const power & p,
7675 const print_myformat & c,
7678 unsigned power_prec = p.precedence();
7679 if (level >= power_prec)
7681 p.op(0).print(c, power_prec);
7683 p.op(1).print(c, power_prec);
7684 if (level >= power_prec)
7690 // install a new print method for power objects
7691 set_print_func<power, print_myformat>(print_power_as_myformat);
7693 // now this prints "1+x**2"
7694 e.print(print_myformat()); cout << endl;
7696 // but the default format is still "1+x^2"
7702 @node Structures, Adding classes, Printing, Extending GiNaC
7703 @c node-name, next, previous, up
7706 If you are doing some very specialized things with GiNaC, or if you just
7707 need some more organized way to store data in your expressions instead of
7708 anonymous lists, you may want to implement your own algebraic classes.
7709 ('algebraic class' means any class directly or indirectly derived from
7710 @code{basic} that can be used in GiNaC expressions).
7712 GiNaC offers two ways of accomplishing this: either by using the
7713 @code{structure<T>} template class, or by rolling your own class from
7714 scratch. This section will discuss the @code{structure<T>} template which
7715 is easier to use but more limited, while the implementation of custom
7716 GiNaC classes is the topic of the next section. However, you may want to
7717 read both sections because many common concepts and member functions are
7718 shared by both concepts, and it will also allow you to decide which approach
7719 is most suited to your needs.
7721 The @code{structure<T>} template, defined in the GiNaC header file
7722 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7723 or @code{class}) into a GiNaC object that can be used in expressions.
7725 @subsection Example: scalar products
7727 Let's suppose that we need a way to handle some kind of abstract scalar
7728 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7729 product class have to store their left and right operands, which can in turn
7730 be arbitrary expressions. Here is a possible way to represent such a
7731 product in a C++ @code{struct}:
7735 #include <ginac/ginac.h>
7736 using namespace std;
7737 using namespace GiNaC;
7743 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7747 The default constructor is required. Now, to make a GiNaC class out of this
7748 data structure, we need only one line:
7751 typedef structure<sprod_s> sprod;
7754 That's it. This line constructs an algebraic class @code{sprod} which
7755 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7756 expressions like any other GiNaC class:
7760 symbol a("a"), b("b");
7761 ex e = sprod(sprod_s(a, b));
7765 Note the difference between @code{sprod} which is the algebraic class, and
7766 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7767 and @code{right} data members. As shown above, an @code{sprod} can be
7768 constructed from an @code{sprod_s} object.
7770 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7771 you could define a little wrapper function like this:
7774 inline ex make_sprod(ex left, ex right)
7776 return sprod(sprod_s(left, right));
7780 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7781 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7782 @code{get_struct()}:
7786 cout << ex_to<sprod>(e)->left << endl;
7788 cout << ex_to<sprod>(e).get_struct().right << endl;
7793 You only have read access to the members of @code{sprod_s}.
7795 The type definition of @code{sprod} is enough to write your own algorithms
7796 that deal with scalar products, for example:
7801 if (is_a<sprod>(p)) @{
7802 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7803 return make_sprod(sp.right, sp.left);
7814 @subsection Structure output
7816 While the @code{sprod} type is useable it still leaves something to be
7817 desired, most notably proper output:
7822 // -> [structure object]
7826 By default, any structure types you define will be printed as
7827 @samp{[structure object]}. To override this you can either specialize the
7828 template's @code{print()} member function, or specify print methods with
7829 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7830 it's not possible to supply class options like @code{print_func<>()} to
7831 structures, so for a self-contained structure type you need to resort to
7832 overriding the @code{print()} function, which is also what we will do here.
7834 The member functions of GiNaC classes are described in more detail in the
7835 next section, but it shouldn't be hard to figure out what's going on here:
7838 void sprod::print(const print_context & c, unsigned level) const
7840 // tree debug output handled by superclass
7841 if (is_a<print_tree>(c))
7842 inherited::print(c, level);
7844 // get the contained sprod_s object
7845 const sprod_s & sp = get_struct();
7847 // print_context::s is a reference to an ostream
7848 c.s << "<" << sp.left << "|" << sp.right << ">";
7852 Now we can print expressions containing scalar products:
7858 cout << swap_sprod(e) << endl;
7863 @subsection Comparing structures
7865 The @code{sprod} class defined so far still has one important drawback: all
7866 scalar products are treated as being equal because GiNaC doesn't know how to
7867 compare objects of type @code{sprod_s}. This can lead to some confusing
7868 and undesired behavior:
7872 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7874 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7875 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7879 To remedy this, we first need to define the operators @code{==} and @code{<}
7880 for objects of type @code{sprod_s}:
7883 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7885 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7888 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7890 return lhs.left.compare(rhs.left) < 0
7891 ? true : lhs.right.compare(rhs.right) < 0;
7895 The ordering established by the @code{<} operator doesn't have to make any
7896 algebraic sense, but it needs to be well defined. Note that we can't use
7897 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7898 in the implementation of these operators because they would construct
7899 GiNaC @code{relational} objects which in the case of @code{<} do not
7900 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7901 decide which one is algebraically 'less').
7903 Next, we need to change our definition of the @code{sprod} type to let
7904 GiNaC know that an ordering relation exists for the embedded objects:
7907 typedef structure<sprod_s, compare_std_less> sprod;
7910 @code{sprod} objects then behave as expected:
7914 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7915 // -> <a|b>-<a^2|b^2>
7916 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7917 // -> <a|b>+<a^2|b^2>
7918 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7920 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7925 The @code{compare_std_less} policy parameter tells GiNaC to use the
7926 @code{std::less} and @code{std::equal_to} functors to compare objects of
7927 type @code{sprod_s}. By default, these functors forward their work to the
7928 standard @code{<} and @code{==} operators, which we have overloaded.
7929 Alternatively, we could have specialized @code{std::less} and
7930 @code{std::equal_to} for class @code{sprod_s}.
7932 GiNaC provides two other comparison policies for @code{structure<T>}
7933 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7934 which does a bit-wise comparison of the contained @code{T} objects.
7935 This should be used with extreme care because it only works reliably with
7936 built-in integral types, and it also compares any padding (filler bytes of
7937 undefined value) that the @code{T} class might have.
7939 @subsection Subexpressions
7941 Our scalar product class has two subexpressions: the left and right
7942 operands. It might be a good idea to make them accessible via the standard
7943 @code{nops()} and @code{op()} methods:
7946 size_t sprod::nops() const
7951 ex sprod::op(size_t i) const
7955 return get_struct().left;
7957 return get_struct().right;
7959 throw std::range_error("sprod::op(): no such operand");
7964 Implementing @code{nops()} and @code{op()} for container types such as
7965 @code{sprod} has two other nice side effects:
7969 @code{has()} works as expected
7971 GiNaC generates better hash keys for the objects (the default implementation
7972 of @code{calchash()} takes subexpressions into account)
7975 @cindex @code{let_op()}
7976 There is a non-const variant of @code{op()} called @code{let_op()} that
7977 allows replacing subexpressions:
7980 ex & sprod::let_op(size_t i)
7982 // every non-const member function must call this
7983 ensure_if_modifiable();
7987 return get_struct().left;
7989 return get_struct().right;
7991 throw std::range_error("sprod::let_op(): no such operand");
7996 Once we have provided @code{let_op()} we also get @code{subs()} and
7997 @code{map()} for free. In fact, every container class that returns a non-null
7998 @code{nops()} value must either implement @code{let_op()} or provide custom
7999 implementations of @code{subs()} and @code{map()}.
8001 In turn, the availability of @code{map()} enables the recursive behavior of a
8002 couple of other default method implementations, in particular @code{evalf()},
8003 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
8004 we probably want to provide our own version of @code{expand()} for scalar
8005 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
8006 This is left as an exercise for the reader.
8008 The @code{structure<T>} template defines many more member functions that
8009 you can override by specialization to customize the behavior of your
8010 structures. You are referred to the next section for a description of
8011 some of these (especially @code{eval()}). There is, however, one topic
8012 that shall be addressed here, as it demonstrates one peculiarity of the
8013 @code{structure<T>} template: archiving.
8015 @subsection Archiving structures
8017 If you don't know how the archiving of GiNaC objects is implemented, you
8018 should first read the next section and then come back here. You're back?
8021 To implement archiving for structures it is not enough to provide
8022 specializations for the @code{archive()} member function and the
8023 unarchiving constructor (the @code{unarchive()} function has a default
8024 implementation). You also need to provide a unique name (as a string literal)
8025 for each structure type you define. This is because in GiNaC archives,
8026 the class of an object is stored as a string, the class name.
8028 By default, this class name (as returned by the @code{class_name()} member
8029 function) is @samp{structure} for all structure classes. This works as long
8030 as you have only defined one structure type, but if you use two or more you
8031 need to provide a different name for each by specializing the
8032 @code{get_class_name()} member function. Here is a sample implementation
8033 for enabling archiving of the scalar product type defined above:
8036 const char *sprod::get_class_name() @{ return "sprod"; @}
8038 void sprod::archive(archive_node & n) const
8040 inherited::archive(n);
8041 n.add_ex("left", get_struct().left);
8042 n.add_ex("right", get_struct().right);
8045 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
8047 n.find_ex("left", get_struct().left, sym_lst);
8048 n.find_ex("right", get_struct().right, sym_lst);
8052 Note that the unarchiving constructor is @code{sprod::structure} and not
8053 @code{sprod::sprod}, and that we don't need to supply an
8054 @code{sprod::unarchive()} function.
8057 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
8058 @c node-name, next, previous, up
8059 @section Adding classes
8061 The @code{structure<T>} template provides an way to extend GiNaC with custom
8062 algebraic classes that is easy to use but has its limitations, the most
8063 severe of which being that you can't add any new member functions to
8064 structures. To be able to do this, you need to write a new class definition
8067 This section will explain how to implement new algebraic classes in GiNaC by
8068 giving the example of a simple 'string' class. After reading this section
8069 you will know how to properly declare a GiNaC class and what the minimum
8070 required member functions are that you have to implement. We only cover the
8071 implementation of a 'leaf' class here (i.e. one that doesn't contain
8072 subexpressions). Creating a container class like, for example, a class
8073 representing tensor products is more involved but this section should give
8074 you enough information so you can consult the source to GiNaC's predefined
8075 classes if you want to implement something more complicated.
8077 @subsection Hierarchy of algebraic classes.
8079 @cindex hierarchy of classes
8080 All algebraic classes (that is, all classes that can appear in expressions)
8081 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
8082 @code{basic *} represents a generic pointer to an algebraic class. Working
8083 with such pointers directly is cumbersome (think of memory management), hence
8084 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
8085 To make such wrapping possible every algebraic class has to implement several
8086 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
8087 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
8088 worry, most of the work is simplified by the following macros (defined
8089 in @file{registrar.h}):
8091 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
8092 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
8093 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
8096 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
8097 required for memory management, visitors, printing, and (un)archiving.
8098 It takes the name of the class and its direct superclass as arguments.
8099 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
8100 the opening brace of the class definition.
8102 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
8103 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
8104 members of a class so that printing and (un)archiving works. The
8105 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
8106 the source (at global scope, of course, not inside a function).
8108 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
8109 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
8110 options, such as custom printing functions.
8112 @subsection A minimalistic example
8114 Now we will start implementing a new class @code{mystring} that allows
8115 placing character strings in algebraic expressions (this is not very useful,
8116 but it's just an example). This class will be a direct subclass of
8117 @code{basic}. You can use this sample implementation as a starting point
8118 for your own classes @footnote{The self-contained source for this example is
8119 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8121 The code snippets given here assume that you have included some header files
8127 #include <stdexcept>
8128 #include <ginac/ginac.h>
8129 using namespace std;
8130 using namespace GiNaC;
8133 Now we can write down the class declaration. The class stores a C++
8134 @code{string} and the user shall be able to construct a @code{mystring}
8135 object from a string:
8138 class mystring : public basic
8140 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8143 mystring(const string & s);
8149 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8152 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8153 for memory management, visitors, printing, and (un)archiving.
8154 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8155 of a class so that printing and (un)archiving works.
8157 Now there are three member functions we have to implement to get a working
8163 @code{mystring()}, the default constructor.
8166 @cindex @code{compare_same_type()}
8167 @code{int compare_same_type(const basic & other)}, which is used internally
8168 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8169 -1, depending on the relative order of this object and the @code{other}
8170 object. If it returns 0, the objects are considered equal.
8171 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8172 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8173 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8174 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8175 must provide a @code{compare_same_type()} function, even those representing
8176 objects for which no reasonable algebraic ordering relationship can be
8180 And, of course, @code{mystring(const string& s)} which is the constructor
8185 Let's proceed step-by-step. The default constructor looks like this:
8188 mystring::mystring() @{ @}
8191 In the default constructor you should set all other member variables to
8192 reasonable default values (we don't need that here since our @code{str}
8193 member gets set to an empty string automatically).
8195 Our @code{compare_same_type()} function uses a provided function to compare
8199 int mystring::compare_same_type(const basic & other) const
8201 const mystring &o = static_cast<const mystring &>(other);
8202 int cmpval = str.compare(o.str);
8205 else if (cmpval < 0)
8212 Although this function takes a @code{basic &}, it will always be a reference
8213 to an object of exactly the same class (objects of different classes are not
8214 comparable), so the cast is safe. If this function returns 0, the two objects
8215 are considered equal (in the sense that @math{A-B=0}), so you should compare
8216 all relevant member variables.
8218 Now the only thing missing is our constructor:
8221 mystring::mystring(const string& s) : str(s) @{ @}
8224 No surprises here. We set the @code{str} member from the argument.
8226 That's it! We now have a minimal working GiNaC class that can store
8227 strings in algebraic expressions. Let's confirm that the RTTI works:
8230 ex e = mystring("Hello, world!");
8231 cout << is_a<mystring>(e) << endl;
8234 cout << ex_to<basic>(e).class_name() << endl;
8238 Obviously it does. Let's see what the expression @code{e} looks like:
8242 // -> [mystring object]
8245 Hm, not exactly what we expect, but of course the @code{mystring} class
8246 doesn't yet know how to print itself. This can be done either by implementing
8247 the @code{print()} member function, or, preferably, by specifying a
8248 @code{print_func<>()} class option. Let's say that we want to print the string
8249 surrounded by double quotes:
8252 class mystring : public basic
8256 void do_print(const print_context & c, unsigned level = 0) const;
8260 void mystring::do_print(const print_context & c, unsigned level) const
8262 // print_context::s is a reference to an ostream
8263 c.s << '\"' << str << '\"';
8267 The @code{level} argument is only required for container classes to
8268 correctly parenthesize the output.
8270 Now we need to tell GiNaC that @code{mystring} objects should use the
8271 @code{do_print()} member function for printing themselves. For this, we
8275 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8281 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8282 print_func<print_context>(&mystring::do_print))
8285 Let's try again to print the expression:
8289 // -> "Hello, world!"
8292 Much better. If we wanted to have @code{mystring} objects displayed in a
8293 different way depending on the output format (default, LaTeX, etc.), we
8294 would have supplied multiple @code{print_func<>()} options with different
8295 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8296 separated by dots. This is similar to the way options are specified for
8297 symbolic functions. @xref{Printing}, for a more in-depth description of the
8298 way expression output is implemented in GiNaC.
8300 The @code{mystring} class can be used in arbitrary expressions:
8303 e += mystring("GiNaC rulez");
8305 // -> "GiNaC rulez"+"Hello, world!"
8308 (GiNaC's automatic term reordering is in effect here), or even
8311 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8313 // -> "One string"^(2*sin(-"Another string"+Pi))
8316 Whether this makes sense is debatable but remember that this is only an
8317 example. At least it allows you to implement your own symbolic algorithms
8320 Note that GiNaC's algebraic rules remain unchanged:
8323 e = mystring("Wow") * mystring("Wow");
8327 e = pow(mystring("First")-mystring("Second"), 2);
8328 cout << e.expand() << endl;
8329 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8332 There's no way to, for example, make GiNaC's @code{add} class perform string
8333 concatenation. You would have to implement this yourself.
8335 @subsection Automatic evaluation
8338 @cindex @code{eval()}
8339 @cindex @code{hold()}
8340 When dealing with objects that are just a little more complicated than the
8341 simple string objects we have implemented, chances are that you will want to
8342 have some automatic simplifications or canonicalizations performed on them.
8343 This is done in the evaluation member function @code{eval()}. Let's say that
8344 we wanted all strings automatically converted to lowercase with
8345 non-alphabetic characters stripped, and empty strings removed:
8348 class mystring : public basic
8352 ex eval() const override;
8356 ex mystring::eval() const
8359 for (size_t i=0; i<str.length(); i++) @{
8361 if (c >= 'A' && c <= 'Z')
8362 new_str += tolower(c);
8363 else if (c >= 'a' && c <= 'z')
8367 if (new_str.length() == 0)
8370 return mystring(new_str).hold();
8374 The @code{hold()} member function sets a flag in the object that prevents
8375 further evaluation. Otherwise we might end up in an endless loop. When you
8376 want to return the object unmodified, use @code{return this->hold();}.
8378 If our class had subobjects, we would have to evaluate them first (unless
8379 they are all of type @code{ex}, which are automatically evaluated). We don't
8380 have any subexpressions in the @code{mystring} class, so we are not concerned
8383 Let's confirm that it works:
8386 ex e = mystring("Hello, world!") + mystring("!?#");
8390 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8395 @subsection Optional member functions
8397 We have implemented only a small set of member functions to make the class
8398 work in the GiNaC framework. There are two functions that are not strictly
8399 required but will make operations with objects of the class more efficient:
8401 @cindex @code{calchash()}
8402 @cindex @code{is_equal_same_type()}
8404 unsigned calchash() const override;
8405 bool is_equal_same_type(const basic & other) const override;
8408 The @code{calchash()} method returns an @code{unsigned} hash value for the
8409 object which will allow GiNaC to compare and canonicalize expressions much
8410 more efficiently. You should consult the implementation of some of the built-in
8411 GiNaC classes for examples of hash functions. The default implementation of
8412 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8413 class and all subexpressions that are accessible via @code{op()}.
8415 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8416 tests for equality without establishing an ordering relation, which is often
8417 faster. The default implementation of @code{is_equal_same_type()} just calls
8418 @code{compare_same_type()} and tests its result for zero.
8420 @subsection Other member functions
8422 For a real algebraic class, there are probably some more functions that you
8423 might want to provide:
8426 bool info(unsigned inf) const override;
8427 ex evalf() const override;
8428 ex series(const relational & r, int order, unsigned options = 0) const override;
8429 ex derivative(const symbol & s) const override;
8432 If your class stores sub-expressions (see the scalar product example in the
8433 previous section) you will probably want to override
8435 @cindex @code{let_op()}
8437 size_t nops() const override;
8438 ex op(size_t i) const override;
8439 ex & let_op(size_t i) override;
8440 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8441 ex map(map_function & f) const override;
8444 @code{let_op()} is a variant of @code{op()} that allows write access. The
8445 default implementations of @code{subs()} and @code{map()} use it, so you have
8446 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8448 You can, of course, also add your own new member functions. Remember
8449 that the RTTI may be used to get information about what kinds of objects
8450 you are dealing with (the position in the class hierarchy) and that you
8451 can always extract the bare object from an @code{ex} by stripping the
8452 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8453 should become a need.
8455 That's it. May the source be with you!
8457 @subsection Upgrading extension classes from older version of GiNaC
8459 GiNaC used to use a custom run time type information system (RTTI). It was
8460 removed from GiNaC. Thus, one needs to rewrite constructors which set
8461 @code{tinfo_key} (which does not exist any more). For example,
8464 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8467 needs to be rewritten as
8470 myclass::myclass() @{@}
8473 @node A comparison with other CAS, Advantages, Adding classes, Top
8474 @c node-name, next, previous, up
8475 @chapter A Comparison With Other CAS
8478 This chapter will give you some information on how GiNaC compares to
8479 other, traditional Computer Algebra Systems, like @emph{Maple},
8480 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8481 disadvantages over these systems.
8484 * Advantages:: Strengths of the GiNaC approach.
8485 * Disadvantages:: Weaknesses of the GiNaC approach.
8486 * Why C++?:: Attractiveness of C++.
8489 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8490 @c node-name, next, previous, up
8493 GiNaC has several advantages over traditional Computer
8494 Algebra Systems, like
8499 familiar language: all common CAS implement their own proprietary
8500 grammar which you have to learn first (and maybe learn again when your
8501 vendor decides to `enhance' it). With GiNaC you can write your program
8502 in common C++, which is standardized.
8506 structured data types: you can build up structured data types using
8507 @code{struct}s or @code{class}es together with STL features instead of
8508 using unnamed lists of lists of lists.
8511 strongly typed: in CAS, you usually have only one kind of variables
8512 which can hold contents of an arbitrary type. This 4GL like feature is
8513 nice for novice programmers, but dangerous.
8516 development tools: powerful development tools exist for C++, like fancy
8517 editors (e.g. with automatic indentation and syntax highlighting),
8518 debuggers, visualization tools, documentation generators@dots{}
8521 modularization: C++ programs can easily be split into modules by
8522 separating interface and implementation.
8525 price: GiNaC is distributed under the GNU Public License which means
8526 that it is free and available with source code. And there are excellent
8527 C++-compilers for free, too.
8530 extendable: you can add your own classes to GiNaC, thus extending it on
8531 a very low level. Compare this to a traditional CAS that you can
8532 usually only extend on a high level by writing in the language defined
8533 by the parser. In particular, it turns out to be almost impossible to
8534 fix bugs in a traditional system.
8537 multiple interfaces: Though real GiNaC programs have to be written in
8538 some editor, then be compiled, linked and executed, there are more ways
8539 to work with the GiNaC engine. Many people want to play with
8540 expressions interactively, as in traditional CASs: The tiny
8541 @command{ginsh} that comes with the distribution exposes many, but not
8542 all, of GiNaC's types to a command line.
8545 seamless integration: it is somewhere between difficult and impossible
8546 to call CAS functions from within a program written in C++ or any other
8547 programming language and vice versa. With GiNaC, your symbolic routines
8548 are part of your program. You can easily call third party libraries,
8549 e.g. for numerical evaluation or graphical interaction. All other
8550 approaches are much more cumbersome: they range from simply ignoring the
8551 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8552 system (i.e. @emph{Yacas}).
8555 efficiency: often large parts of a program do not need symbolic
8556 calculations at all. Why use large integers for loop variables or
8557 arbitrary precision arithmetics where @code{int} and @code{double} are
8558 sufficient? For pure symbolic applications, GiNaC is comparable in
8559 speed with other CAS.
8564 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8565 @c node-name, next, previous, up
8566 @section Disadvantages
8568 Of course it also has some disadvantages:
8573 advanced features: GiNaC cannot compete with a program like
8574 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8575 which grows since 1981 by the work of dozens of programmers, with
8576 respect to mathematical features. Integration,
8577 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8578 not planned for the near future).
8581 portability: While the GiNaC library itself is designed to avoid any
8582 platform dependent features (it should compile on any ANSI compliant C++
8583 compiler), the currently used version of the CLN library (fast large
8584 integer and arbitrary precision arithmetics) can only by compiled
8585 without hassle on systems with the C++ compiler from the GNU Compiler
8586 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8587 macros to let the compiler gather all static initializations, which
8588 works for GNU C++ only. Feel free to contact the authors in case you
8589 really believe that you need to use a different compiler. We have
8590 occasionally used other compilers and may be able to give you advice.}
8591 GiNaC uses recent language features like explicit constructors, mutable
8592 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8598 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8599 @c node-name, next, previous, up
8602 Why did we choose to implement GiNaC in C++ instead of Java or any other
8603 language? C++ is not perfect: type checking is not strict (casting is
8604 possible), separation between interface and implementation is not
8605 complete, object oriented design is not enforced. The main reason is
8606 the often scolded feature of operator overloading in C++. While it may
8607 be true that operating on classes with a @code{+} operator is rarely
8608 meaningful, it is perfectly suited for algebraic expressions. Writing
8609 @math{3x+5y} as @code{3*x+5*y} instead of
8610 @code{x.times(3).plus(y.times(5))} looks much more natural.
8611 Furthermore, the main developers are more familiar with C++ than with
8612 any other programming language.
8615 @node Internal structures, Expressions are reference counted, Why C++? , Top
8616 @c node-name, next, previous, up
8617 @appendix Internal structures
8620 * Expressions are reference counted::
8621 * Internal representation of products and sums::
8624 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8625 @c node-name, next, previous, up
8626 @appendixsection Expressions are reference counted
8628 @cindex reference counting
8629 @cindex copy-on-write
8630 @cindex garbage collection
8631 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8632 where the counter belongs to the algebraic objects derived from class
8633 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8634 which @code{ex} contains an instance. If you understood that, you can safely
8635 skip the rest of this passage.
8637 Expressions are extremely light-weight since internally they work like
8638 handles to the actual representation. They really hold nothing more
8639 than a pointer to some other object. What this means in practice is
8640 that whenever you create two @code{ex} and set the second equal to the
8641 first no copying process is involved. Instead, the copying takes place
8642 as soon as you try to change the second. Consider the simple sequence
8647 #include <ginac/ginac.h>
8648 using namespace std;
8649 using namespace GiNaC;
8653 symbol x("x"), y("y"), z("z");
8656 e1 = sin(x + 2*y) + 3*z + 41;
8657 e2 = e1; // e2 points to same object as e1
8658 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8659 e2 += 1; // e2 is copied into a new object
8660 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8664 The line @code{e2 = e1;} creates a second expression pointing to the
8665 object held already by @code{e1}. The time involved for this operation
8666 is therefore constant, no matter how large @code{e1} was. Actual
8667 copying, however, must take place in the line @code{e2 += 1;} because
8668 @code{e1} and @code{e2} are not handles for the same object any more.
8669 This concept is called @dfn{copy-on-write semantics}. It increases
8670 performance considerably whenever one object occurs multiple times and
8671 represents a simple garbage collection scheme because when an @code{ex}
8672 runs out of scope its destructor checks whether other expressions handle
8673 the object it points to too and deletes the object from memory if that
8674 turns out not to be the case. A slightly less trivial example of
8675 differentiation using the chain-rule should make clear how powerful this
8680 symbol x("x"), y("y");
8684 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8685 cout << e1 << endl // prints x+3*y
8686 << e2 << endl // prints (x+3*y)^3
8687 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8691 Here, @code{e1} will actually be referenced three times while @code{e2}
8692 will be referenced two times. When the power of an expression is built,
8693 that expression needs not be copied. Likewise, since the derivative of
8694 a power of an expression can be easily expressed in terms of that
8695 expression, no copying of @code{e1} is involved when @code{e3} is
8696 constructed. So, when @code{e3} is constructed it will print as
8697 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8698 holds a reference to @code{e2} and the factor in front is just
8701 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8702 semantics. When you insert an expression into a second expression, the
8703 result behaves exactly as if the contents of the first expression were
8704 inserted. But it may be useful to remember that this is not what
8705 happens. Knowing this will enable you to write much more efficient
8706 code. If you still have an uncertain feeling with copy-on-write
8707 semantics, we recommend you have a look at the
8708 @uref{https://isocpp.org/faq, C++-FAQ's} chapter on memory management.
8709 It covers this issue and presents an implementation which is pretty
8710 close to the one in GiNaC.
8713 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8714 @c node-name, next, previous, up
8715 @appendixsection Internal representation of products and sums
8717 @cindex representation
8720 @cindex @code{power}
8721 Although it should be completely transparent for the user of
8722 GiNaC a short discussion of this topic helps to understand the sources
8723 and also explain performance to a large degree. Consider the
8724 unexpanded symbolic expression
8726 $2d^3 \left( 4a + 5b - 3 \right)$
8729 @math{2*d^3*(4*a+5*b-3)}
8731 which could naively be represented by a tree of linear containers for
8732 addition and multiplication, one container for exponentiation with base
8733 and exponent and some atomic leaves of symbols and numbers in this
8743 @cindex pair-wise representation
8744 However, doing so results in a rather deeply nested tree which will
8745 quickly become inefficient to manipulate. We can improve on this by
8746 representing the sum as a sequence of terms, each one being a pair of a
8747 purely numeric multiplicative coefficient and its rest. In the same
8748 spirit we can store the multiplication as a sequence of terms, each
8749 having a numeric exponent and a possibly complicated base, the tree
8750 becomes much more flat:
8759 The number @code{3} above the symbol @code{d} shows that @code{mul}
8760 objects are treated similarly where the coefficients are interpreted as
8761 @emph{exponents} now. Addition of sums of terms or multiplication of
8762 products with numerical exponents can be coded to be very efficient with
8763 such a pair-wise representation. Internally, this handling is performed
8764 by most CAS in this way. It typically speeds up manipulations by an
8765 order of magnitude. The overall multiplicative factor @code{2} and the
8766 additive term @code{-3} look somewhat out of place in this
8767 representation, however, since they are still carrying a trivial
8768 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8769 this is avoided by adding a field that carries an overall numeric
8770 coefficient. This results in the realistic picture of internal
8773 $2d^3 \left( 4a + 5b - 3 \right)$:
8776 @math{2*d^3*(4*a+5*b-3)}:
8787 This also allows for a better handling of numeric radicals, since
8788 @code{sqrt(2)} can now be carried along calculations. Now it should be
8789 clear, why both classes @code{add} and @code{mul} are derived from the
8790 same abstract class: the data representation is the same, only the
8791 semantics differs. In the class hierarchy, methods for polynomial
8792 expansion and the like are reimplemented for @code{add} and @code{mul},
8793 but the data structure is inherited from @code{expairseq}.
8796 @node Package tools, Configure script options, Internal representation of products and sums, Top
8797 @c node-name, next, previous, up
8798 @appendix Package tools
8800 If you are creating a software package that uses the GiNaC library,
8801 setting the correct command line options for the compiler and linker can
8802 be difficult. The @command{pkg-config} utility makes this process
8803 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8804 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8805 program use @footnote{If GiNaC is installed into some non-standard
8806 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8807 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8809 g++ -o simple simple.cpp `pkg-config --cflags --libs ginac`
8812 This command line might expand to (for example):
8814 g++ -o simple simple.cpp -lginac -lcln
8817 Not only is the form using @command{pkg-config} easier to type, it will
8818 work on any system, no matter how GiNaC was configured.
8820 For packages configured using GNU automake, @command{pkg-config} also
8821 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8822 checking for libraries
8825 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8826 [@var{ACTION-IF-FOUND}],
8827 [@var{ACTION-IF-NOT-FOUND}])
8835 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8836 either found in the default @command{pkg-config} search path, or from
8837 the environment variable @env{PKG_CONFIG_PATH}.
8840 Tests the installed libraries to make sure that their version
8841 is later than @var{MINIMUM-VERSION}.
8844 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8845 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8846 variable to the output of @command{pkg-config --libs ginac}, and calls
8847 @samp{AC_SUBST()} for these variables so they can be used in generated
8848 makefiles, and then executes @var{ACTION-IF-FOUND}.
8851 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8856 * Configure script options:: Configuring a package that uses GiNaC
8857 * Example package:: Example of a package using GiNaC
8861 @node Configure script options, Example package, Package tools, Package tools
8862 @c node-name, next, previous, up
8863 @appendixsection Configuring a package that uses GiNaC
8865 The directory where the GiNaC libraries are installed needs
8866 to be found by your system's dynamic linkers (both compile- and run-time
8867 ones). See the documentation of your system linker for details. Also
8868 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8869 @xref{pkg-config, ,pkg-config, *manpages*}.
8871 The short summary below describes how to do this on a GNU/Linux
8874 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8875 the linkers where to find the library one should
8879 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8881 # echo PREFIX/lib >> /etc/ld.so.conf
8886 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8888 $ export LD_LIBRARY_PATH=PREFIX/lib
8889 $ export LD_RUN_PATH=PREFIX/lib
8893 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8897 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8901 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8902 set the @env{PKG_CONFIG_PATH} environment variable:
8904 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8907 Finally, run the @command{configure} script
8912 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8914 @node Example package, Bibliography, Configure script options, Package tools
8915 @c node-name, next, previous, up
8916 @appendixsection Example of a package using GiNaC
8918 The following shows how to build a simple package using automake
8919 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8923 #include <ginac/ginac.h>
8927 GiNaC::symbol x("x");
8928 GiNaC::ex a = GiNaC::sin(x);
8929 std::cout << "Derivative of " << a
8930 << " is " << a.diff(x) << std::endl;
8935 You should first read the introductory portions of the automake
8936 Manual, if you are not already familiar with it.
8938 Two files are needed, @file{configure.ac}, which is used to build the
8942 dnl Process this file with autoreconf to produce a configure script.
8943 AC_INIT([simple], 1.0.0, bogus@@example.net)
8944 AC_CONFIG_SRCDIR(simple.cpp)
8945 AM_INIT_AUTOMAKE([foreign 1.8])
8951 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8956 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8957 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8958 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8960 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8962 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8964 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8965 installed software in a non-standard prefix.
8967 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8968 and SIMPLE_LIBS to avoid the need to call pkg-config.
8969 See the pkg-config man page for more details.
8972 And the @file{Makefile.am}, which will be used to build the Makefile.
8975 ## Process this file with automake to produce Makefile.in
8976 bin_PROGRAMS = simple
8977 simple_SOURCES = simple.cpp
8978 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8979 simple_LDADD = $(SIMPLE_LIBS)
8982 This @file{Makefile.am}, says that we are building a single executable,
8983 from a single source file @file{simple.cpp}. Since every program
8984 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8985 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8986 more flexible to specify libraries and complier options on a per-program
8989 To try this example out, create a new directory and add the three
8992 Now execute the following command:
8998 You now have a package that can be built in the normal fashion
9007 @node Bibliography, Concept index, Example package, Top
9008 @c node-name, next, previous, up
9009 @appendix Bibliography
9014 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
9017 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
9020 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
9023 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
9026 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
9027 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
9030 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
9031 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
9032 Academic Press, London
9035 @cite{Computer Algebra Systems - A Practical Guide},
9036 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
9039 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
9040 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
9043 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
9044 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
9047 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
9052 @node Concept index, , Bibliography, Top
9053 @c node-name, next, previous, up
9054 @unnumbered Concept index