1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
18 @dircategory Mathematics
20 * ginac: (ginac). C++ library for symbolic computation.
24 This is a tutorial that documents GiNaC @value{VERSION}, an open
25 framework for symbolic computation within the C++ programming language.
27 Copyright (C) 1999-2020 Johannes Gutenberg University Mainz, Germany
29 Permission is granted to make and distribute verbatim copies of
30 this manual provided the copyright notice and this permission notice
31 are preserved on all copies.
34 Permission is granted to process this file through TeX and print the
35 results, provided the printed document carries copying permission
36 notice identical to this one except for the removal of this paragraph
39 Permission is granted to copy and distribute modified versions of this
40 manual under the conditions for verbatim copying, provided that the entire
41 resulting derived work is distributed under the terms of a permission
42 notice identical to this one.
46 @c finalout prevents ugly black rectangles on overfull hbox lines
48 @title GiNaC @value{VERSION}
49 @subtitle An open framework for symbolic computation within the C++ programming language
50 @subtitle @value{UPDATED}
51 @author @uref{https://www.ginac.de}
54 @vskip 0pt plus 1filll
55 Copyright @copyright{} 1999-2020 Johannes Gutenberg University Mainz, Germany
57 Permission is granted to make and distribute verbatim copies of
58 this manual provided the copyright notice and this permission notice
59 are preserved on all copies.
61 Permission is granted to copy and distribute modified versions of this
62 manual under the conditions for verbatim copying, provided that the entire
63 resulting derived work is distributed under the terms of a permission
64 notice identical to this one.
73 @node Top, Introduction, (dir), (dir)
74 @c node-name, next, previous, up
77 This is a tutorial that documents GiNaC @value{VERSION}, an open
78 framework for symbolic computation within the C++ programming language.
81 * Introduction:: GiNaC's purpose.
82 * A tour of GiNaC:: A quick tour of the library.
83 * Installation:: How to install the package.
84 * Basic concepts:: Description of fundamental classes.
85 * Methods and functions:: Algorithms for symbolic manipulations.
86 * Extending GiNaC:: How to extend the library.
87 * A comparison with other CAS:: Compares GiNaC to traditional CAS.
88 * Internal structures:: Description of some internal structures.
89 * Package tools:: Configuring packages to work with GiNaC.
95 @node Introduction, A tour of GiNaC, Top, Top
96 @c node-name, next, previous, up
98 @cindex history of GiNaC
100 The motivation behind GiNaC derives from the observation that most
101 present day computer algebra systems (CAS) are linguistically and
102 semantically impoverished. Although they are quite powerful tools for
103 learning math and solving particular problems they lack modern
104 linguistic structures that allow for the creation of large-scale
105 projects. GiNaC is an attempt to overcome this situation by extending a
106 well established and standardized computer language (C++) by some
107 fundamental symbolic capabilities, thus allowing for integrated systems
108 that embed symbolic manipulations together with more established areas
109 of computer science (like computation-intense numeric applications,
110 graphical interfaces, etc.) under one roof.
112 The particular problem that led to the writing of the GiNaC framework is
113 still a very active field of research, namely the calculation of higher
114 order corrections to elementary particle interactions. There,
115 theoretical physicists are interested in matching present day theories
116 against experiments taking place at particle accelerators. The
117 computations involved are so complex they call for a combined symbolical
118 and numerical approach. This turned out to be quite difficult to
119 accomplish with the present day CAS we have worked with so far and so we
120 tried to fill the gap by writing GiNaC. But of course its applications
121 are in no way restricted to theoretical physics.
123 This tutorial is intended for the novice user who is new to GiNaC but
124 already has some background in C++ programming. However, since a
125 hand-made documentation like this one is difficult to keep in sync with
126 the development, the actual documentation is inside the sources in the
127 form of comments. That documentation may be parsed by one of the many
128 Javadoc-like documentation systems. If you fail at generating it you
129 may access it from @uref{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{} 1999-2020 Johannes Gutenberg
139 University Mainz, Germany.
141 This program is free software; you can redistribute it and/or
142 modify it under the terms of the GNU General Public License as
143 published by the Free Software Foundation; either version 2 of the
144 License, or (at your option) any later version.
146 This program is distributed in the hope that it will be useful, but
147 WITHOUT ANY WARRANTY; without even the implied warranty of
148 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
149 General Public License for more details.
151 You should have received a copy of the GNU General Public License
152 along with this program; see the file COPYING. If not, write to the
153 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
157 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
158 @c node-name, next, previous, up
159 @chapter A Tour of GiNaC
161 This quick tour of GiNaC wants to arise your interest in the
162 subsequent chapters by showing off a bit. Please excuse us if it
163 leaves many open questions.
166 * How to use it from within C++:: Two simple examples.
167 * What it can do for you:: A Tour of GiNaC's features.
171 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
172 @c node-name, next, previous, up
173 @section How to use it from within C++
175 The GiNaC open framework for symbolic computation within the C++ programming
176 language does not try to define a language of its own as conventional
177 CAS do. Instead, it extends the capabilities of C++ by symbolic
178 manipulations. Here is how to generate and print a simple (and rather
179 pointless) bivariate polynomial with some large coefficients:
183 #include <ginac/ginac.h>
185 using namespace GiNaC;
189 symbol x("x"), y("y");
192 for (int i=0; i<3; ++i)
193 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
195 cout << poly << endl;
200 Assuming the file is called @file{hello.cc}, on our system we can compile
201 and run it like this:
204 $ c++ hello.cc -o hello -lginac -lcln
206 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
209 (@xref{Package tools}, for tools that help you when creating a software
210 package that uses GiNaC.)
212 @cindex Hermite polynomial
213 Next, there is a more meaningful C++ program that calls a function which
214 generates Hermite polynomials in a specified free variable.
218 #include <ginac/ginac.h>
220 using namespace GiNaC;
222 ex HermitePoly(const symbol & x, int n)
224 ex HKer=exp(-pow(x, 2));
225 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
226 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
233 for (int i=0; i<6; ++i)
234 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
240 When run, this will type out
246 H_3(z) == -12*z+8*z^3
247 H_4(z) == -48*z^2+16*z^4+12
248 H_5(z) == 120*z-160*z^3+32*z^5
251 This method of generating the coefficients is of course far from optimal
252 for production purposes.
254 In order to show some more examples of what GiNaC can do we will now use
255 the @command{ginsh}, a simple GiNaC interactive shell that provides a
256 convenient window into GiNaC's capabilities.
259 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
260 @c node-name, next, previous, up
261 @section What it can do for you
263 @cindex @command{ginsh}
264 After invoking @command{ginsh} one can test and experiment with GiNaC's
265 features much like in other Computer Algebra Systems except that it does
266 not provide programming constructs like loops or conditionals. For a
267 concise description of the @command{ginsh} syntax we refer to its
268 accompanied man page. Suffice to say that assignments and comparisons in
269 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
272 It can manipulate arbitrary precision integers in a very fast way.
273 Rational numbers are automatically converted to fractions of coprime
278 369988485035126972924700782451696644186473100389722973815184405301748249
280 123329495011708990974900260817232214728824366796574324605061468433916083
287 Exact numbers are always retained as exact numbers and only evaluated as
288 floating point numbers if requested. For instance, with numeric
289 radicals is dealt pretty much as with symbols. Products of sums of them
293 > expand((1+a^(1/5)-a^(2/5))^3);
294 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
295 > expand((1+3^(1/5)-3^(2/5))^3);
297 > evalf((1+3^(1/5)-3^(2/5))^3);
298 0.33408977534118624228
301 The function @code{evalf} that was used above converts any number in
302 GiNaC's expressions into floating point numbers. This can be done to
303 arbitrary predefined accuracy:
307 0.14285714285714285714
311 0.1428571428571428571428571428571428571428571428571428571428571428571428
312 5714285714285714285714285714285714285
315 Exact numbers other than rationals that can be manipulated in GiNaC
316 include predefined constants like Archimedes' @code{Pi}. They can both
317 be used in symbolic manipulations (as an exact number) as well as in
318 numeric expressions (as an inexact number):
324 9.869604401089358619+x
328 11.869604401089358619
331 Built-in functions evaluate immediately to exact numbers if
332 this is possible. Conversions that can be safely performed are done
333 immediately; conversions that are not generally valid are not done:
344 (Note that converting the last input to @code{x} would allow one to
345 conclude that @code{42*Pi} is equal to @code{0}.)
347 Linear equation systems can be solved along with basic linear
348 algebra manipulations over symbolic expressions. In C++ GiNaC offers
349 a matrix class for this purpose but we can see what it can do using
350 @command{ginsh}'s bracket notation to type them in:
353 > lsolve(a+x*y==z,x);
355 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
357 > M = [ [1, 3], [-3, 2] ];
361 > charpoly(M,lambda);
363 > A = [ [1, 1], [2, -1] ];
366 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
369 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
370 > evalm(B^(2^12345));
371 [[1,0,0],[0,1,0],[0,0,1]]
374 Multivariate polynomials and rational functions may be expanded,
375 collected, 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. @inforef{Introduction, , cln}, 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
1869 unknown variables. But the most common usage of objects of this class
1870 is rather inconspicuous in statements of the form @code{if
1871 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1872 conversion from @code{relational} to @code{bool} takes place. Note,
1873 however, that @code{==} here does not perform any simplifications, hence
1874 @code{expand()} must be called explicitly.
1876 @node Integrals, Matrices, Relations, Basic concepts
1877 @c node-name, next, previous, up
1879 @cindex @code{integral} (class)
1881 An object of class @dfn{integral} can be used to hold a symbolic integral.
1882 If you want to symbolically represent the integral of @code{x*x} from 0 to
1883 1, you would write this as
1885 integral(x, 0, 1, x*x)
1887 The first argument is the integration variable. It should be noted that
1888 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1889 fact, it can only integrate polynomials. An expression containing integrals
1890 can be evaluated symbolically by calling the
1894 method on it. Numerical evaluation is available by calling the
1898 method on an expression containing the integral. This will only evaluate
1899 integrals into a number if @code{subs}ing the integration variable by a
1900 number in the fourth argument of an integral and then @code{evalf}ing the
1901 result always results in a number. Of course, also the boundaries of the
1902 integration domain must @code{evalf} into numbers. It should be noted that
1903 trying to @code{evalf} a function with discontinuities in the integration
1904 domain is not recommended. The accuracy of the numeric evaluation of
1905 integrals is determined by the static member variable
1907 ex integral::relative_integration_error
1909 of the class @code{integral}. The default value of this is 10^-8.
1910 The integration works by halving the interval of integration, until numeric
1911 stability of the answer indicates that the requested accuracy has been
1912 reached. The maximum depth of the halving can be set via the static member
1915 int integral::max_integration_level
1917 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1918 return the integral unevaluated. The function that performs the numerical
1919 evaluation, is also available as
1921 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1924 This function will throw an exception if the maximum depth is exceeded. The
1925 last parameter of the function is optional and defaults to the
1926 @code{relative_integration_error}. To make sure that we do not do too
1927 much work if an expression contains the same integral multiple times,
1928 a lookup table is used.
1930 If you know that an expression holds an integral, you can get the
1931 integration variable, the left boundary, right boundary and integrand by
1932 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1933 @code{.op(3)}. Differentiating integrals with respect to variables works
1934 as expected. Note that it makes no sense to differentiate an integral
1935 with respect to the integration variable.
1937 @node Matrices, Indexed objects, Integrals, Basic concepts
1938 @c node-name, next, previous, up
1940 @cindex @code{matrix} (class)
1942 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1943 matrix with @math{m} rows and @math{n} columns are accessed with two
1944 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1945 second one in the range 0@dots{}@math{n-1}.
1947 There are a couple of ways to construct matrices, with or without preset
1948 elements. The constructor
1951 matrix::matrix(unsigned r, unsigned c);
1954 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1957 The easiest way to create a matrix is using an initializer list of
1958 initializer lists, all of the same size:
1962 matrix m = @{@{1, -a@},
1967 You can also specify the elements as a (flat) list with
1970 matrix::matrix(unsigned r, unsigned c, const lst & l);
1975 @cindex @code{lst_to_matrix()}
1977 ex lst_to_matrix(const lst & l);
1980 constructs a matrix from a list of lists, each list representing a matrix row.
1982 There is also a set of functions for creating some special types of
1985 @cindex @code{diag_matrix()}
1986 @cindex @code{unit_matrix()}
1987 @cindex @code{symbolic_matrix()}
1989 ex diag_matrix(const lst & l);
1990 ex diag_matrix(initializer_list<ex> l);
1991 ex unit_matrix(unsigned x);
1992 ex unit_matrix(unsigned r, unsigned c);
1993 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1994 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1995 const string & tex_base_name);
1998 @code{diag_matrix()} constructs a square diagonal matrix given the diagonal
1999 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
2000 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
2001 matrix filled with newly generated symbols made of the specified base name
2002 and the position of each element in the matrix.
2004 Matrices often arise by omitting elements of another matrix. For
2005 instance, the submatrix @code{S} of a matrix @code{M} takes a
2006 rectangular block from @code{M}. The reduced matrix @code{R} is defined
2007 by removing one row and one column from a matrix @code{M}. (The
2008 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
2009 can be used for computing the inverse using Cramer's rule.)
2011 @cindex @code{sub_matrix()}
2012 @cindex @code{reduced_matrix()}
2014 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
2015 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
2018 The function @code{sub_matrix()} takes a row offset @code{r} and a
2019 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
2020 columns. The function @code{reduced_matrix()} has two integer arguments
2021 that specify which row and column to remove:
2025 matrix m = @{@{11, 12, 13@},
2028 cout << reduced_matrix(m, 1, 1) << endl;
2029 // -> [[11,13],[31,33]]
2030 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2031 // -> [[22,23],[32,33]]
2035 Matrix elements can be accessed and set using the parenthesis (function call)
2039 const ex & matrix::operator()(unsigned r, unsigned c) const;
2040 ex & matrix::operator()(unsigned r, unsigned c);
2043 It is also possible to access the matrix elements in a linear fashion with
2044 the @code{op()} method. But C++-style subscripting with square brackets
2045 @samp{[]} is not available.
2047 Here are a couple of examples for constructing matrices:
2051 symbol a("a"), b("b");
2053 matrix M = @{@{a, 0@},
2064 cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
2067 cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
2070 cout << diag_matrix(lst@{a, b@}) << endl;
2073 cout << unit_matrix(3) << endl;
2074 // -> [[1,0,0],[0,1,0],[0,0,1]]
2076 cout << symbolic_matrix(2, 3, "x") << endl;
2077 // -> [[x00,x01,x02],[x10,x11,x12]]
2081 @cindex @code{is_zero_matrix()}
2082 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2083 all entries of the matrix are zeros. There is also method
2084 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2085 expression is zero or a zero matrix.
2087 @cindex @code{transpose()}
2088 There are three ways to do arithmetic with matrices. The first (and most
2089 direct one) is to use the methods provided by the @code{matrix} class:
2092 matrix matrix::add(const matrix & other) const;
2093 matrix matrix::sub(const matrix & other) const;
2094 matrix matrix::mul(const matrix & other) const;
2095 matrix matrix::mul_scalar(const ex & other) const;
2096 matrix matrix::pow(const ex & expn) const;
2097 matrix matrix::transpose() const;
2100 All of these methods return the result as a new matrix object. Here is an
2101 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2106 matrix A = @{@{ 1, 2@},
2108 matrix B = @{@{-1, 0@},
2110 matrix C = @{@{ 8, 4@},
2113 matrix result = A.mul(B).sub(C.mul_scalar(2));
2114 cout << result << endl;
2115 // -> [[-13,-6],[1,2]]
2120 @cindex @code{evalm()}
2121 The second (and probably the most natural) way is to construct an expression
2122 containing matrices with the usual arithmetic operators and @code{pow()}.
2123 For efficiency reasons, expressions with sums, products and powers of
2124 matrices are not automatically evaluated in GiNaC. You have to call the
2128 ex ex::evalm() const;
2131 to obtain the result:
2138 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2139 cout << e.evalm() << endl;
2140 // -> [[-13,-6],[1,2]]
2145 The non-commutativity of the product @code{A*B} in this example is
2146 automatically recognized by GiNaC. There is no need to use a special
2147 operator here. @xref{Non-commutative objects}, for more information about
2148 dealing with non-commutative expressions.
2150 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2151 to perform the arithmetic:
2156 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2157 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2159 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2160 cout << e.simplify_indexed() << endl;
2161 // -> [[-13,-6],[1,2]].i.j
2165 Using indices is most useful when working with rectangular matrices and
2166 one-dimensional vectors because you don't have to worry about having to
2167 transpose matrices before multiplying them. @xref{Indexed objects}, for
2168 more information about using matrices with indices, and about indices in
2171 The @code{matrix} class provides a couple of additional methods for
2172 computing determinants, traces, characteristic polynomials and ranks:
2174 @cindex @code{determinant()}
2175 @cindex @code{trace()}
2176 @cindex @code{charpoly()}
2177 @cindex @code{rank()}
2179 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2180 ex matrix::trace() const;
2181 ex matrix::charpoly(const ex & lambda) const;
2182 unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;
2185 The optional @samp{algo} argument of @code{determinant()} and @code{rank()}
2186 functions allows to select between different algorithms for calculating the
2187 determinant and rank respectively. The asymptotic speed (as parametrized
2188 by the matrix size) can greatly differ between those algorithms, depending
2189 on the nature of the matrix' entries. The possible values are defined in
2190 the @file{flags.h} header file. By default, GiNaC uses a heuristic to
2191 automatically select an algorithm that is likely (but not guaranteed)
2192 to give the result most quickly.
2194 @cindex @code{solve()}
2195 Linear systems can be solved with:
2198 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2199 unsigned algo=solve_algo::automatic) const;
2202 Assuming the matrix object this method is applied on is an @code{m}
2203 times @code{n} matrix, then @code{vars} must be a @code{n} times
2204 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2205 times @code{p} matrix. The returned matrix then has dimension @code{n}
2206 times @code{p} and in the case of an underdetermined system will still
2207 contain some of the indeterminates from @code{vars}. If the system is
2208 overdetermined, an exception is thrown.
2210 @cindex @code{inverse()} (matrix)
2211 To invert a matrix, use the method:
2214 matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
2217 The @samp{algo} argument is optional. If given, it must be one of
2218 @code{solve_algo} defined in @file{flags.h}.
2220 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2221 @c node-name, next, previous, up
2222 @section Indexed objects
2224 GiNaC allows you to handle expressions containing general indexed objects in
2225 arbitrary spaces. It is also able to canonicalize and simplify such
2226 expressions and perform symbolic dummy index summations. There are a number
2227 of predefined indexed objects provided, like delta and metric tensors.
2229 There are few restrictions placed on indexed objects and their indices and
2230 it is easy to construct nonsense expressions, but our intention is to
2231 provide a general framework that allows you to implement algorithms with
2232 indexed quantities, getting in the way as little as possible.
2234 @cindex @code{idx} (class)
2235 @cindex @code{indexed} (class)
2236 @subsection Indexed quantities and their indices
2238 Indexed expressions in GiNaC are constructed of two special types of objects,
2239 @dfn{index objects} and @dfn{indexed objects}.
2243 @cindex contravariant
2246 @item Index objects are of class @code{idx} or a subclass. Every index has
2247 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2248 the index lives in) which can both be arbitrary expressions but are usually
2249 a number or a simple symbol. In addition, indices of class @code{varidx} have
2250 a @dfn{variance} (they can be co- or contravariant), and indices of class
2251 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2253 @item Indexed objects are of class @code{indexed} or a subclass. They
2254 contain a @dfn{base expression} (which is the expression being indexed), and
2255 one or more indices.
2259 @strong{Please notice:} when printing expressions, covariant indices and indices
2260 without variance are denoted @samp{.i} while contravariant indices are
2261 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2262 value. In the following, we are going to use that notation in the text so
2263 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2264 not visible in the output.
2266 A simple example shall illustrate the concepts:
2270 #include <ginac/ginac.h>
2271 using namespace std;
2272 using namespace GiNaC;
2276 symbol i_sym("i"), j_sym("j");
2277 idx i(i_sym, 3), j(j_sym, 3);
2280 cout << indexed(A, i, j) << endl;
2282 cout << index_dimensions << indexed(A, i, j) << endl;
2284 cout << dflt; // reset cout to default output format (dimensions hidden)
2288 The @code{idx} constructor takes two arguments, the index value and the
2289 index dimension. First we define two index objects, @code{i} and @code{j},
2290 both with the numeric dimension 3. The value of the index @code{i} is the
2291 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2292 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2293 construct an expression containing one indexed object, @samp{A.i.j}. It has
2294 the symbol @code{A} as its base expression and the two indices @code{i} and
2297 The dimensions of indices are normally not visible in the output, but one
2298 can request them to be printed with the @code{index_dimensions} manipulator,
2301 Note the difference between the indices @code{i} and @code{j} which are of
2302 class @code{idx}, and the index values which are the symbols @code{i_sym}
2303 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2304 or numbers but must be index objects. For example, the following is not
2305 correct and will raise an exception:
2308 symbol i("i"), j("j");
2309 e = indexed(A, i, j); // ERROR: indices must be of type idx
2312 You can have multiple indexed objects in an expression, index values can
2313 be numeric, and index dimensions symbolic:
2317 symbol B("B"), dim("dim");
2318 cout << 4 * indexed(A, i)
2319 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2324 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2325 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2326 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2327 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2328 @code{simplify_indexed()} for that, see below).
2330 In fact, base expressions, index values and index dimensions can be
2331 arbitrary expressions:
2335 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2340 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2341 get an error message from this but you will probably not be able to do
2342 anything useful with it.
2344 @cindex @code{get_value()}
2345 @cindex @code{get_dim()}
2349 ex idx::get_value();
2353 return the value and dimension of an @code{idx} object. If you have an index
2354 in an expression, such as returned by calling @code{.op()} on an indexed
2355 object, you can get a reference to the @code{idx} object with the function
2356 @code{ex_to<idx>()} on the expression.
2358 There are also the methods
2361 bool idx::is_numeric();
2362 bool idx::is_symbolic();
2363 bool idx::is_dim_numeric();
2364 bool idx::is_dim_symbolic();
2367 for checking whether the value and dimension are numeric or symbolic
2368 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2369 about expressions}) returns information about the index value.
2371 @cindex @code{varidx} (class)
2372 If you need co- and contravariant indices, use the @code{varidx} class:
2376 symbol mu_sym("mu"), nu_sym("nu");
2377 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2378 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2380 cout << indexed(A, mu, nu) << endl;
2382 cout << indexed(A, mu_co, nu) << endl;
2384 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2389 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2390 co- or contravariant. The default is a contravariant (upper) index, but
2391 this can be overridden by supplying a third argument to the @code{varidx}
2392 constructor. The two methods
2395 bool varidx::is_covariant();
2396 bool varidx::is_contravariant();
2399 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2400 to get the object reference from an expression). There's also the very useful
2404 ex varidx::toggle_variance();
2407 which makes a new index with the same value and dimension but the opposite
2408 variance. By using it you only have to define the index once.
2410 @cindex @code{spinidx} (class)
2411 The @code{spinidx} class provides dotted and undotted variant indices, as
2412 used in the Weyl-van-der-Waerden spinor formalism:
2416 symbol K("K"), C_sym("C"), D_sym("D");
2417 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2418 // contravariant, undotted
2419 spinidx C_co(C_sym, 2, true); // covariant index
2420 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2421 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2423 cout << indexed(K, C, D) << endl;
2425 cout << indexed(K, C_co, D_dot) << endl;
2427 cout << indexed(K, D_co_dot, D) << endl;
2432 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2433 dotted or undotted. The default is undotted but this can be overridden by
2434 supplying a fourth argument to the @code{spinidx} constructor. The two
2438 bool spinidx::is_dotted();
2439 bool spinidx::is_undotted();
2442 allow you to check whether or not a @code{spinidx} object is dotted (use
2443 @code{ex_to<spinidx>()} to get the object reference from an expression).
2444 Finally, the two methods
2447 ex spinidx::toggle_dot();
2448 ex spinidx::toggle_variance_dot();
2451 create a new index with the same value and dimension but opposite dottedness
2452 and the same or opposite variance.
2454 @subsection Substituting indices
2456 @cindex @code{subs()}
2457 Sometimes you will want to substitute one symbolic index with another
2458 symbolic or numeric index, for example when calculating one specific element
2459 of a tensor expression. This is done with the @code{.subs()} method, as it
2460 is done for symbols (see @ref{Substituting expressions}).
2462 You have two possibilities here. You can either substitute the whole index
2463 by another index or expression:
2467 ex e = indexed(A, mu_co);
2468 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2469 // -> A.mu becomes A~nu
2470 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2471 // -> A.mu becomes A~0
2472 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2473 // -> A.mu becomes A.0
2477 The third example shows that trying to replace an index with something that
2478 is not an index will substitute the index value instead.
2480 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2485 ex e = indexed(A, mu_co);
2486 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2487 // -> A.mu becomes A.nu
2488 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2489 // -> A.mu becomes A.0
2493 As you see, with the second method only the value of the index will get
2494 substituted. Its other properties, including its dimension, remain unchanged.
2495 If you want to change the dimension of an index you have to substitute the
2496 whole index by another one with the new dimension.
2498 Finally, substituting the base expression of an indexed object works as
2503 ex e = indexed(A, mu_co);
2504 cout << e << " becomes " << e.subs(A == A+B) << endl;
2505 // -> A.mu becomes (B+A).mu
2509 @subsection Symmetries
2510 @cindex @code{symmetry} (class)
2511 @cindex @code{sy_none()}
2512 @cindex @code{sy_symm()}
2513 @cindex @code{sy_anti()}
2514 @cindex @code{sy_cycl()}
2516 Indexed objects can have certain symmetry properties with respect to their
2517 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2518 that is constructed with the helper functions
2521 symmetry sy_none(...);
2522 symmetry sy_symm(...);
2523 symmetry sy_anti(...);
2524 symmetry sy_cycl(...);
2527 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2528 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2529 represents a cyclic symmetry. Each of these functions accepts up to four
2530 arguments which can be either symmetry objects themselves or unsigned integer
2531 numbers that represent an index position (counting from 0). A symmetry
2532 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2533 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2536 Here are some examples of symmetry definitions:
2541 e = indexed(A, i, j);
2542 e = indexed(A, sy_none(), i, j); // equivalent
2543 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2545 // Symmetric in all three indices:
2546 e = indexed(A, sy_symm(), i, j, k);
2547 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2548 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2549 // different canonical order
2551 // Symmetric in the first two indices only:
2552 e = indexed(A, sy_symm(0, 1), i, j, k);
2553 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2555 // Antisymmetric in the first and last index only (index ranges need not
2557 e = indexed(A, sy_anti(0, 2), i, j, k);
2558 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2560 // An example of a mixed symmetry: antisymmetric in the first two and
2561 // last two indices, symmetric when swapping the first and last index
2562 // pairs (like the Riemann curvature tensor):
2563 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2565 // Cyclic symmetry in all three indices:
2566 e = indexed(A, sy_cycl(), i, j, k);
2567 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2569 // The following examples are invalid constructions that will throw
2570 // an exception at run time.
2572 // An index may not appear multiple times:
2573 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2574 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2576 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2577 // same number of indices:
2578 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2580 // And of course, you cannot specify indices which are not there:
2581 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2585 If you need to specify more than four indices, you have to use the
2586 @code{.add()} method of the @code{symmetry} class. For example, to specify
2587 full symmetry in the first six indices you would write
2588 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2590 If an indexed object has a symmetry, GiNaC will automatically bring the
2591 indices into a canonical order which allows for some immediate simplifications:
2595 cout << indexed(A, sy_symm(), i, j)
2596 + indexed(A, sy_symm(), j, i) << endl;
2598 cout << indexed(B, sy_anti(), i, j)
2599 + indexed(B, sy_anti(), j, i) << endl;
2601 cout << indexed(B, sy_anti(), i, j, k)
2602 - indexed(B, sy_anti(), j, k, i) << endl;
2607 @cindex @code{get_free_indices()}
2609 @subsection Dummy indices
2611 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2612 that a summation over the index range is implied. Symbolic indices which are
2613 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2614 dummy nor free indices.
2616 To be recognized as a dummy index pair, the two indices must be of the same
2617 class and their value must be the same single symbol (an index like
2618 @samp{2*n+1} is never a dummy index). If the indices are of class
2619 @code{varidx} they must also be of opposite variance; if they are of class
2620 @code{spinidx} they must be both dotted or both undotted.
2622 The method @code{.get_free_indices()} returns a vector containing the free
2623 indices of an expression. It also checks that the free indices of the terms
2624 of a sum are consistent:
2628 symbol A("A"), B("B"), C("C");
2630 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2631 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2633 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2634 cout << exprseq(e.get_free_indices()) << endl;
2636 // 'j' and 'l' are dummy indices
2638 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2639 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2641 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2642 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2643 cout << exprseq(e.get_free_indices()) << endl;
2645 // 'nu' is a dummy index, but 'sigma' is not
2647 e = indexed(A, mu, mu);
2648 cout << exprseq(e.get_free_indices()) << endl;
2650 // 'mu' is not a dummy index because it appears twice with the same
2653 e = indexed(A, mu, nu) + 42;
2654 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2655 // this will throw an exception:
2656 // "add::get_free_indices: inconsistent indices in sum"
2660 @cindex @code{expand_dummy_sum()}
2661 A dummy index summation like
2668 can be expanded for indices with numeric
2669 dimensions (e.g. 3) into the explicit sum like
2671 $a_1b^1+a_2b^2+a_3b^3 $.
2674 a.1 b~1 + a.2 b~2 + a.3 b~3.
2676 This is performed by the function
2679 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2682 which takes an expression @code{e} and returns the expanded sum for all
2683 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2684 is set to @code{true} then all substitutions are made by @code{idx} class
2685 indices, i.e. without variance. In this case the above sum
2694 $a_1b_1+a_2b_2+a_3b_3 $.
2697 a.1 b.1 + a.2 b.2 + a.3 b.3.
2701 @cindex @code{simplify_indexed()}
2702 @subsection Simplifying indexed expressions
2704 In addition to the few automatic simplifications that GiNaC performs on
2705 indexed expressions (such as re-ordering the indices of symmetric tensors
2706 and calculating traces and convolutions of matrices and predefined tensors)
2710 ex ex::simplify_indexed();
2711 ex ex::simplify_indexed(const scalar_products & sp);
2714 that performs some more expensive operations:
2717 @item it checks the consistency of free indices in sums in the same way
2718 @code{get_free_indices()} does
2719 @item it tries to give dummy indices that appear in different terms of a sum
2720 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2721 @item it (symbolically) calculates all possible dummy index summations/contractions
2722 with the predefined tensors (this will be explained in more detail in the
2724 @item it detects contractions that vanish for symmetry reasons, for example
2725 the contraction of a symmetric and a totally antisymmetric tensor
2726 @item as a special case of dummy index summation, it can replace scalar products
2727 of two tensors with a user-defined value
2730 The last point is done with the help of the @code{scalar_products} class
2731 which is used to store scalar products with known values (this is not an
2732 arithmetic class, you just pass it to @code{simplify_indexed()}):
2736 symbol A("A"), B("B"), C("C"), i_sym("i");
2740 sp.add(A, B, 0); // A and B are orthogonal
2741 sp.add(A, C, 0); // A and C are orthogonal
2742 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2744 e = indexed(A + B, i) * indexed(A + C, i);
2746 // -> (B+A).i*(A+C).i
2748 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2754 The @code{scalar_products} object @code{sp} acts as a storage for the
2755 scalar products added to it with the @code{.add()} method. This method
2756 takes three arguments: the two expressions of which the scalar product is
2757 taken, and the expression to replace it with.
2759 @cindex @code{expand()}
2760 The example above also illustrates a feature of the @code{expand()} method:
2761 if passed the @code{expand_indexed} option it will distribute indices
2762 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2764 @cindex @code{tensor} (class)
2765 @subsection Predefined tensors
2767 Some frequently used special tensors such as the delta, epsilon and metric
2768 tensors are predefined in GiNaC. They have special properties when
2769 contracted with other tensor expressions and some of them have constant
2770 matrix representations (they will evaluate to a number when numeric
2771 indices are specified).
2773 @cindex @code{delta_tensor()}
2774 @subsubsection Delta tensor
2776 The delta tensor takes two indices, is symmetric and has the matrix
2777 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2778 @code{delta_tensor()}:
2782 symbol A("A"), B("B");
2784 idx i(symbol("i"), 3), j(symbol("j"), 3),
2785 k(symbol("k"), 3), l(symbol("l"), 3);
2787 ex e = indexed(A, i, j) * indexed(B, k, l)
2788 * delta_tensor(i, k) * delta_tensor(j, l);
2789 cout << e.simplify_indexed() << endl;
2792 cout << delta_tensor(i, i) << endl;
2797 @cindex @code{metric_tensor()}
2798 @subsubsection General metric tensor
2800 The function @code{metric_tensor()} creates a general symmetric metric
2801 tensor with two indices that can be used to raise/lower tensor indices. The
2802 metric tensor is denoted as @samp{g} in the output and if its indices are of
2803 mixed variance it is automatically replaced by a delta tensor:
2809 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2811 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2812 cout << e.simplify_indexed() << endl;
2815 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2816 cout << e.simplify_indexed() << endl;
2819 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2820 * metric_tensor(nu, rho);
2821 cout << e.simplify_indexed() << endl;
2824 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2825 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2826 + indexed(A, mu.toggle_variance(), rho));
2827 cout << e.simplify_indexed() << endl;
2832 @cindex @code{lorentz_g()}
2833 @subsubsection Minkowski metric tensor
2835 The Minkowski metric tensor is a special metric tensor with a constant
2836 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2837 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2838 It is created with the function @code{lorentz_g()} (although it is output as
2843 varidx mu(symbol("mu"), 4);
2845 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2846 * lorentz_g(mu, varidx(0, 4)); // negative signature
2847 cout << e.simplify_indexed() << endl;
2850 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2851 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2852 cout << e.simplify_indexed() << endl;
2857 @cindex @code{spinor_metric()}
2858 @subsubsection Spinor metric tensor
2860 The function @code{spinor_metric()} creates an antisymmetric tensor with
2861 two indices that is used to raise/lower indices of 2-component spinors.
2862 It is output as @samp{eps}:
2868 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2869 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2871 e = spinor_metric(A, B) * indexed(psi, B_co);
2872 cout << e.simplify_indexed() << endl;
2875 e = spinor_metric(A, B) * indexed(psi, A_co);
2876 cout << e.simplify_indexed() << endl;
2879 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2880 cout << e.simplify_indexed() << endl;
2883 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2884 cout << e.simplify_indexed() << endl;
2887 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2888 cout << e.simplify_indexed() << endl;
2891 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2892 cout << e.simplify_indexed() << endl;
2897 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2899 @cindex @code{epsilon_tensor()}
2900 @cindex @code{lorentz_eps()}
2901 @subsubsection Epsilon tensor
2903 The epsilon tensor is totally antisymmetric, its number of indices is equal
2904 to the dimension of the index space (the indices must all be of the same
2905 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2906 defined to be 1. Its behavior with indices that have a variance also
2907 depends on the signature of the metric. Epsilon tensors are output as
2910 There are three functions defined to create epsilon tensors in 2, 3 and 4
2914 ex epsilon_tensor(const ex & i1, const ex & i2);
2915 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2916 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2917 bool pos_sig = false);
2920 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2921 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2922 Minkowski space (the last @code{bool} argument specifies whether the metric
2923 has negative or positive signature, as in the case of the Minkowski metric
2928 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2929 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2930 e = lorentz_eps(mu, nu, rho, sig) *
2931 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2932 cout << simplify_indexed(e) << endl;
2933 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2935 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2936 symbol A("A"), B("B");
2937 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2938 cout << simplify_indexed(e) << endl;
2939 // -> -B.k*A.j*eps.i.k.j
2940 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2941 cout << simplify_indexed(e) << endl;
2946 @subsection Linear algebra
2948 The @code{matrix} class can be used with indices to do some simple linear
2949 algebra (linear combinations and products of vectors and matrices, traces
2950 and scalar products):
2954 idx i(symbol("i"), 2), j(symbol("j"), 2);
2955 symbol x("x"), y("y");
2957 // A is a 2x2 matrix, X is a 2x1 vector
2958 matrix A = @{@{1, 2@},
2960 matrix X = @{@{x, y@}@};
2962 cout << indexed(A, i, i) << endl;
2965 ex e = indexed(A, i, j) * indexed(X, j);
2966 cout << e.simplify_indexed() << endl;
2967 // -> [[2*y+x],[4*y+3*x]].i
2969 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2970 cout << e.simplify_indexed() << endl;
2971 // -> [[3*y+3*x,6*y+2*x]].j
2975 You can of course obtain the same results with the @code{matrix::add()},
2976 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2977 but with indices you don't have to worry about transposing matrices.
2979 Matrix indices always start at 0 and their dimension must match the number
2980 of rows/columns of the matrix. Matrices with one row or one column are
2981 vectors and can have one or two indices (it doesn't matter whether it's a
2982 row or a column vector). Other matrices must have two indices.
2984 You should be careful when using indices with variance on matrices. GiNaC
2985 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2986 @samp{F.mu.nu} are different matrices. In this case you should use only
2987 one form for @samp{F} and explicitly multiply it with a matrix representation
2988 of the metric tensor.
2991 @node Non-commutative objects, Methods and functions, Indexed objects, Basic concepts
2992 @c node-name, next, previous, up
2993 @section Non-commutative objects
2995 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2996 non-commutative objects are built-in which are mostly of use in high energy
3000 @item Clifford (Dirac) algebra (class @code{clifford})
3001 @item su(3) Lie algebra (class @code{color})
3002 @item Matrices (unindexed) (class @code{matrix})
3005 The @code{clifford} and @code{color} classes are subclasses of
3006 @code{indexed} because the elements of these algebras usually carry
3007 indices. The @code{matrix} class is described in more detail in
3010 Unlike most computer algebra systems, GiNaC does not primarily provide an
3011 operator (often denoted @samp{&*}) for representing inert products of
3012 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
3013 classes of objects involved, and non-commutative products are formed with
3014 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
3015 figuring out by itself which objects commutate and will group the factors
3016 by their class. Consider this example:
3020 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3021 idx a(symbol("a"), 8), b(symbol("b"), 8);
3022 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
3024 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3028 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3029 groups the non-commutative factors (the gammas and the su(3) generators)
3030 together while preserving the order of factors within each class (because
3031 Clifford objects commutate with color objects). The resulting expression is a
3032 @emph{commutative} product with two factors that are themselves non-commutative
3033 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3034 parentheses are placed around the non-commutative products in the output.
3036 @cindex @code{ncmul} (class)
3037 Non-commutative products are internally represented by objects of the class
3038 @code{ncmul}, as opposed to commutative products which are handled by the
3039 @code{mul} class. You will normally not have to worry about this distinction,
3042 The advantage of this approach is that you never have to worry about using
3043 (or forgetting to use) a special operator when constructing non-commutative
3044 expressions. Also, non-commutative products in GiNaC are more intelligent
3045 than in other computer algebra systems; they can, for example, automatically
3046 canonicalize themselves according to rules specified in the implementation
3047 of the non-commutative classes. The drawback is that to work with other than
3048 the built-in algebras you have to implement new classes yourself. Both
3049 symbols and user-defined functions can be specified as being non-commutative.
3050 For symbols, this is done by subclassing class symbol; for functions,
3051 by explicitly setting the return type (@pxref{Symbolic functions}).
3053 @cindex @code{return_type()}
3054 @cindex @code{return_type_tinfo()}
3055 Information about the commutativity of an object or expression can be
3056 obtained with the two member functions
3059 unsigned ex::return_type() const;
3060 return_type_t ex::return_type_tinfo() const;
3063 The @code{return_type()} function returns one of three values (defined in
3064 the header file @file{flags.h}), corresponding to three categories of
3065 expressions in GiNaC:
3068 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3069 classes are of this kind.
3070 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3071 certain class of non-commutative objects which can be determined with the
3072 @code{return_type_tinfo()} method. Expressions of this category commutate
3073 with everything except @code{noncommutative} expressions of the same
3075 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3076 of non-commutative objects of different classes. Expressions of this
3077 category don't commutate with any other @code{noncommutative} or
3078 @code{noncommutative_composite} expressions.
3081 The @code{return_type_tinfo()} method returns an object of type
3082 @code{return_type_t} that contains information about the type of the expression
3083 and, if given, its representation label (see section on dirac gamma matrices for
3084 more details). The objects of type @code{return_type_t} can be tested for
3085 equality to test whether two expressions belong to the same category and
3086 therefore may not commute.
3088 Here are a couple of examples:
3091 @multitable @columnfractions .6 .4
3092 @item @strong{Expression} @tab @strong{@code{return_type()}}
3093 @item @code{42} @tab @code{commutative}
3094 @item @code{2*x-y} @tab @code{commutative}
3095 @item @code{dirac_ONE()} @tab @code{noncommutative}
3096 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
3097 @item @code{2*color_T(a)} @tab @code{noncommutative}
3098 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
3102 A last note: With the exception of matrices, positive integer powers of
3103 non-commutative objects are automatically expanded in GiNaC. For example,
3104 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3105 non-commutative expressions).
3108 @cindex @code{clifford} (class)
3109 @subsection Clifford algebra
3112 Clifford algebras are supported in two flavours: Dirac gamma
3113 matrices (more physical) and generic Clifford algebras (more
3116 @cindex @code{dirac_gamma()}
3117 @subsubsection Dirac gamma matrices
3118 Dirac gamma matrices (note that GiNaC doesn't treat them
3119 as matrices) are designated as @samp{gamma~mu} and satisfy
3120 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3121 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3122 constructed by the function
3125 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3128 which takes two arguments: the index and a @dfn{representation label} in the
3129 range 0 to 255 which is used to distinguish elements of different Clifford
3130 algebras (this is also called a @dfn{spin line index}). Gammas with different
3131 labels commutate with each other. The dimension of the index can be 4 or (in
3132 the framework of dimensional regularization) any symbolic value. Spinor
3133 indices on Dirac gammas are not supported in GiNaC.
3135 @cindex @code{dirac_ONE()}
3136 The unity element of a Clifford algebra is constructed by
3139 ex dirac_ONE(unsigned char rl = 0);
3142 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3143 multiples of the unity element, even though it's customary to omit it.
3144 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3145 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3146 GiNaC will complain and/or produce incorrect results.
3148 @cindex @code{dirac_gamma5()}
3149 There is a special element @samp{gamma5} that commutates with all other
3150 gammas, has a unit square, and in 4 dimensions equals
3151 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3154 ex dirac_gamma5(unsigned char rl = 0);
3157 @cindex @code{dirac_gammaL()}
3158 @cindex @code{dirac_gammaR()}
3159 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3160 objects, constructed by
3163 ex dirac_gammaL(unsigned char rl = 0);
3164 ex dirac_gammaR(unsigned char rl = 0);
3167 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3168 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3170 @cindex @code{dirac_slash()}
3171 Finally, the function
3174 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3177 creates a term that represents a contraction of @samp{e} with the Dirac
3178 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3179 with a unique index whose dimension is given by the @code{dim} argument).
3180 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3182 In products of dirac gammas, superfluous unity elements are automatically
3183 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3184 and @samp{gammaR} are moved to the front.
3186 The @code{simplify_indexed()} function performs contractions in gamma strings,
3192 symbol a("a"), b("b"), D("D");
3193 varidx mu(symbol("mu"), D);
3194 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3195 * dirac_gamma(mu.toggle_variance());
3197 // -> gamma~mu*a\*gamma.mu
3198 e = e.simplify_indexed();
3201 cout << e.subs(D == 4) << endl;
3207 @cindex @code{dirac_trace()}
3208 To calculate the trace of an expression containing strings of Dirac gammas
3209 you use one of the functions
3212 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3213 const ex & trONE = 4);
3214 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3215 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3218 These functions take the trace over all gammas in the specified set @code{rls}
3219 or list @code{rll} of representation labels, or the single label @code{rl};
3220 gammas with other labels are left standing. The last argument to
3221 @code{dirac_trace()} is the value to be returned for the trace of the unity
3222 element, which defaults to 4.
3224 The @code{dirac_trace()} function is a linear functional that is equal to the
3225 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3226 functional is not cyclic in
3232 dimensions when acting on
3233 expressions containing @samp{gamma5}, so it's not a proper trace. This
3234 @samp{gamma5} scheme is described in greater detail in the article
3235 @cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
3237 The value of the trace itself is also usually different in 4 and in
3248 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3249 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3250 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3251 cout << dirac_trace(e).simplify_indexed() << endl;
3258 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3259 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3260 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3261 cout << dirac_trace(e).simplify_indexed() << endl;
3262 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3266 Here is an example for using @code{dirac_trace()} to compute a value that
3267 appears in the calculation of the one-loop vacuum polarization amplitude in
3272 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3273 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3276 sp.add(l, l, pow(l, 2));
3277 sp.add(l, q, ldotq);
3279 ex e = dirac_gamma(mu) *
3280 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3281 dirac_gamma(mu.toggle_variance()) *
3282 (dirac_slash(l, D) + m * dirac_ONE());
3283 e = dirac_trace(e).simplify_indexed(sp);
3284 e = e.collect(lst@{l, ldotq, m@});
3286 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3290 The @code{canonicalize_clifford()} function reorders all gamma products that
3291 appear in an expression to a canonical (but not necessarily simple) form.
3292 You can use this to compare two expressions or for further simplifications:
3296 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3297 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3299 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3301 e = canonicalize_clifford(e);
3303 // -> 2*ONE*eta~mu~nu
3307 @cindex @code{clifford_unit()}
3308 @subsubsection A generic Clifford algebra
3310 A generic Clifford algebra, i.e. a
3316 dimensional algebra with
3323 satisfying the identities
3325 $e_i e_j + e_j e_i = M(i, j) + M(j, i)$
3328 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3330 for some bilinear form (@code{metric})
3331 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3332 and contain symbolic entries. Such generators are created by the
3336 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
3339 where @code{mu} should be a @code{idx} (or descendant) class object
3340 indexing the generators.
3341 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3342 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3343 object. In fact, any expression either with two free indices or without
3344 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3345 object with two newly created indices with @code{metr} as its
3346 @code{op(0)} will be used.
3347 Optional parameter @code{rl} allows to distinguish different
3348 Clifford algebras, which will commute with each other.
3350 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3351 something very close to @code{dirac_gamma(mu)}, although
3352 @code{dirac_gamma} have more efficient simplification mechanism.
3353 @cindex @code{get_metric()}
3354 Also, the object created by @code{clifford_unit(mu, minkmetric())} is
3355 not aware about the symmetry of its metric, see the start of the previous
3356 paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
3357 specifies as follows:
3360 clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
3363 The method @code{clifford::get_metric()} returns a metric defining this
3366 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3367 the Clifford algebra units with a call like that
3370 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3373 since this may yield some further automatic simplifications. Again, for a
3374 metric defined through a @code{matrix} such a symmetry is detected
3377 Individual generators of a Clifford algebra can be accessed in several
3383 idx i(symbol("i"), 4);
3385 ex M = diag_matrix(lst@{1, -1, 0, s@});
3386 ex e = clifford_unit(i, M);
3387 ex e0 = e.subs(i == 0);
3388 ex e1 = e.subs(i == 1);
3389 ex e2 = e.subs(i == 2);
3390 ex e3 = e.subs(i == 3);
3395 will produce four anti-commuting generators of a Clifford algebra with properties
3397 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3400 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3401 @code{pow(e3, 2) = s}.
3404 @cindex @code{lst_to_clifford()}
3405 A similar effect can be achieved from the function
3408 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3409 unsigned char rl = 0);
3410 ex lst_to_clifford(const ex & v, const ex & e);
3413 which converts a list or vector
3415 $v = (v^0, v^1, ..., v^n)$
3418 @samp{v = (v~0, v~1, ..., v~n)}
3423 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3426 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3429 directly supplied in the second form of the procedure. In the first form
3430 the Clifford unit @samp{e.k} is generated by the call of
3431 @code{clifford_unit(mu, metr, rl)}.
3432 @cindex pseudo-vector
3433 If the number of components supplied
3434 by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
3435 1 then function @code{lst_to_clifford()} uses the following
3436 pseudo-vector representation:
3438 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3441 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3444 The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
3449 idx i(symbol("i"), 4);
3451 ex M = diag_matrix(@{1, -1, 0, s@});
3452 ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
3453 ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
3454 ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
3455 ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
3460 @cindex @code{clifford_to_lst()}
3461 There is the inverse function
3464 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3467 which takes an expression @code{e} and tries to find a list
3469 $v = (v^0, v^1, ..., v^n)$
3472 @samp{v = (v~0, v~1, ..., v~n)}
3474 such that the expression is either vector
3476 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3479 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3483 $v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
3486 @samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
3488 with respect to the given Clifford units @code{c}. Here none of the
3489 @samp{v~k} should contain Clifford units @code{c} (of course, this
3490 may be impossible). This function can use an @code{algebraic} method
3491 (default) or a symbolic one. With the @code{algebraic} method the
3492 @samp{v~k} are calculated as
3494 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3497 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3499 is zero or is not @code{numeric} for some @samp{k}
3500 then the method will be automatically changed to symbolic. The same effect
3501 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3503 @cindex @code{clifford_prime()}
3504 @cindex @code{clifford_star()}
3505 @cindex @code{clifford_bar()}
3506 There are several functions for (anti-)automorphisms of Clifford algebras:
3509 ex clifford_prime(const ex & e)
3510 inline ex clifford_star(const ex & e)
3511 inline ex clifford_bar(const ex & e)
3514 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3515 changes signs of all Clifford units in the expression. The reversion
3516 of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
3517 units in any product. Finally the main anti-automorphism
3518 of a Clifford algebra @code{clifford_bar()} is the composition of the
3519 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3520 in a product. These functions correspond to the notations
3535 used in Clifford algebra textbooks.
3537 @cindex @code{clifford_norm()}
3541 ex clifford_norm(const ex & e);
3544 @cindex @code{clifford_inverse()}
3545 calculates the norm of a Clifford number from the expression
3547 $||e||^2 = e\overline{e}$.
3550 @code{||e||^2 = e \bar@{e@}}
3552 The inverse of a Clifford expression is returned by the function
3555 ex clifford_inverse(const ex & e);
3558 which calculates it as
3560 $e^{-1} = \overline{e}/||e||^2$.
3563 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3572 then an exception is raised.
3574 @cindex @code{remove_dirac_ONE()}
3575 If a Clifford number happens to be a factor of
3576 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3577 expression by the function
3580 ex remove_dirac_ONE(const ex & e);
3583 @cindex @code{canonicalize_clifford()}
3584 The function @code{canonicalize_clifford()} works for a
3585 generic Clifford algebra in a similar way as for Dirac gammas.
3587 The next provided function is
3589 @cindex @code{clifford_moebius_map()}
3591 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3592 const ex & d, const ex & v, const ex & G,
3593 unsigned char rl = 0);
3594 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3595 unsigned char rl = 0);
3598 It takes a list or vector @code{v} and makes the Moebius (conformal or
3599 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3600 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3601 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3602 indexed object, tensormetric, matrix or a Clifford unit, in the later
3603 case the optional parameter @code{rl} is ignored even if supplied.
3604 Depending from the type of @code{v} the returned value of this function
3605 is either a vector or a list holding vector's components.
3607 @cindex @code{clifford_max_label()}
3608 Finally the function
3611 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3614 can detect a presence of Clifford objects in the expression @code{e}: if
3615 such objects are found it returns the maximal
3616 @code{representation_label} of them, otherwise @code{-1}. The optional
3617 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3618 be ignored during the search.
3620 LaTeX output for Clifford units looks like
3621 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3622 @code{representation_label} and @code{\nu} is the index of the
3623 corresponding unit. This provides a flexible typesetting with a suitable
3624 definition of the @code{\clifford} command. For example, the definition
3626 \newcommand@{\clifford@}[1][]@{@}
3628 typesets all Clifford units identically, while the alternative definition
3630 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3632 prints units with @code{representation_label=0} as
3639 with @code{representation_label=1} as
3646 and with @code{representation_label=2} as
3654 @cindex @code{color} (class)
3655 @subsection Color algebra
3657 @cindex @code{color_T()}
3658 For computations in quantum chromodynamics, GiNaC implements the base elements
3659 and structure constants of the su(3) Lie algebra (color algebra). The base
3660 elements @math{T_a} are constructed by the function
3663 ex color_T(const ex & a, unsigned char rl = 0);
3666 which takes two arguments: the index and a @dfn{representation label} in the
3667 range 0 to 255 which is used to distinguish elements of different color
3668 algebras. Objects with different labels commutate with each other. The
3669 dimension of the index must be exactly 8 and it should be of class @code{idx},
3672 @cindex @code{color_ONE()}
3673 The unity element of a color algebra is constructed by
3676 ex color_ONE(unsigned char rl = 0);
3679 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3680 multiples of the unity element, even though it's customary to omit it.
3681 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3682 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3683 GiNaC may produce incorrect results.
3685 @cindex @code{color_d()}
3686 @cindex @code{color_f()}
3690 ex color_d(const ex & a, const ex & b, const ex & c);
3691 ex color_f(const ex & a, const ex & b, const ex & c);
3694 create the symmetric and antisymmetric structure constants @math{d_abc} and
3695 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3696 and @math{[T_a, T_b] = i f_abc T_c}.
3698 These functions evaluate to their numerical values,
3699 if you supply numeric indices to them. The index values should be in
3700 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3701 goes along better with the notations used in physical literature.
3703 @cindex @code{color_h()}
3704 There's an additional function
3707 ex color_h(const ex & a, const ex & b, const ex & c);
3710 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3712 The function @code{simplify_indexed()} performs some simplifications on
3713 expressions containing color objects:
3718 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3719 k(symbol("k"), 8), l(symbol("l"), 8);
3721 e = color_d(a, b, l) * color_f(a, b, k);
3722 cout << e.simplify_indexed() << endl;
3725 e = color_d(a, b, l) * color_d(a, b, k);
3726 cout << e.simplify_indexed() << endl;
3729 e = color_f(l, a, b) * color_f(a, b, k);
3730 cout << e.simplify_indexed() << endl;
3733 e = color_h(a, b, c) * color_h(a, b, c);
3734 cout << e.simplify_indexed() << endl;
3737 e = color_h(a, b, c) * color_T(b) * color_T(c);
3738 cout << e.simplify_indexed() << endl;
3741 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3742 cout << e.simplify_indexed() << endl;
3745 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3746 cout << e.simplify_indexed() << endl;
3747 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3751 @cindex @code{color_trace()}
3752 To calculate the trace of an expression containing color objects you use one
3756 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3757 ex color_trace(const ex & e, const lst & rll);
3758 ex color_trace(const ex & e, unsigned char rl = 0);
3761 These functions take the trace over all color @samp{T} objects in the
3762 specified set @code{rls} or list @code{rll} of representation labels, or the
3763 single label @code{rl}; @samp{T}s with other labels are left standing. For
3768 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3770 // -> -I*f.a.c.b+d.a.c.b
3775 @node Methods and functions, Information about expressions, Non-commutative objects, Top
3776 @c node-name, next, previous, up
3777 @chapter Methods and functions
3780 In this chapter the most important algorithms provided by GiNaC will be
3781 described. Some of them are implemented as functions on expressions,
3782 others are implemented as methods provided by expression objects. If
3783 they are methods, there exists a wrapper function around it, so you can
3784 alternatively call it in a functional way as shown in the simple
3789 cout << "As method: " << sin(1).evalf() << endl;
3790 cout << "As function: " << evalf(sin(1)) << endl;
3794 @cindex @code{subs()}
3795 The general rule is that wherever methods accept one or more parameters
3796 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3797 wrapper accepts is the same but preceded by the object to act on
3798 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3799 most natural one in an OO model but it may lead to confusion for MapleV
3800 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3801 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3802 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3803 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3804 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3805 here. Also, users of MuPAD will in most cases feel more comfortable
3806 with GiNaC's convention. All function wrappers are implemented
3807 as simple inline functions which just call the corresponding method and
3808 are only provided for users uncomfortable with OO who are dead set to
3809 avoid method invocations. Generally, nested function wrappers are much
3810 harder to read than a sequence of methods and should therefore be
3811 avoided if possible. On the other hand, not everything in GiNaC is a
3812 method on class @code{ex} and sometimes calling a function cannot be
3816 * Information about expressions::
3817 * Numerical evaluation::
3818 * Substituting expressions::
3819 * Pattern matching and advanced substitutions::
3820 * Applying a function on subexpressions::
3821 * Visitors and tree traversal::
3822 * Polynomial arithmetic:: Working with polynomials.
3823 * Rational expressions:: Working with rational functions.
3824 * Symbolic differentiation::
3825 * Series expansion:: Taylor and Laurent expansion.
3827 * Built-in functions:: List of predefined mathematical functions.
3828 * Multiple polylogarithms::
3829 * Complex expressions::
3830 * Solving linear systems of equations::
3831 * Input/output:: Input and output of expressions.
3835 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3836 @c node-name, next, previous, up
3837 @section Getting information about expressions
3839 @subsection Checking expression types
3840 @cindex @code{is_a<@dots{}>()}
3841 @cindex @code{is_exactly_a<@dots{}>()}
3842 @cindex @code{ex_to<@dots{}>()}
3843 @cindex Converting @code{ex} to other classes
3844 @cindex @code{info()}
3845 @cindex @code{return_type()}
3846 @cindex @code{return_type_tinfo()}
3848 Sometimes it's useful to check whether a given expression is a plain number,
3849 a sum, a polynomial with integer coefficients, or of some other specific type.
3850 GiNaC provides a couple of functions for this:
3853 bool is_a<T>(const ex & e);
3854 bool is_exactly_a<T>(const ex & e);
3855 bool ex::info(unsigned flag);
3856 unsigned ex::return_type() const;
3857 return_type_t ex::return_type_tinfo() const;
3860 When the test made by @code{is_a<T>()} returns true, it is safe to call
3861 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3862 class names (@xref{The class hierarchy}, for a list of all classes). For
3863 example, assuming @code{e} is an @code{ex}:
3868 if (is_a<numeric>(e))
3869 numeric n = ex_to<numeric>(e);
3874 @code{is_a<T>(e)} allows you to check whether the top-level object of
3875 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3876 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3877 e.g., for checking whether an expression is a number, a sum, or a product:
3884 is_a<numeric>(e1); // true
3885 is_a<numeric>(e2); // false
3886 is_a<add>(e1); // false
3887 is_a<add>(e2); // true
3888 is_a<mul>(e1); // false
3889 is_a<mul>(e2); // false
3893 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3894 top-level object of an expression @samp{e} is an instance of the GiNaC
3895 class @samp{T}, not including parent classes.
3897 The @code{info()} method is used for checking certain attributes of
3898 expressions. The possible values for the @code{flag} argument are defined
3899 in @file{ginac/flags.h}, the most important being explained in the following
3903 @multitable @columnfractions .30 .70
3904 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3905 @item @code{numeric}
3906 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3908 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3909 @item @code{rational}
3910 @tab @dots{}an exact rational number (integers are rational, too)
3911 @item @code{integer}
3912 @tab @dots{}a (non-complex) integer
3913 @item @code{crational}
3914 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3915 @item @code{cinteger}
3916 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3917 @item @code{positive}
3918 @tab @dots{}not complex and greater than 0
3919 @item @code{negative}
3920 @tab @dots{}not complex and less than 0
3921 @item @code{nonnegative}
3922 @tab @dots{}not complex and greater than or equal to 0
3924 @tab @dots{}an integer greater than 0
3926 @tab @dots{}an integer less than 0
3927 @item @code{nonnegint}
3928 @tab @dots{}an integer greater than or equal to 0
3930 @tab @dots{}an even integer
3932 @tab @dots{}an odd integer
3934 @tab @dots{}a prime integer (probabilistic primality test)
3935 @item @code{relation}
3936 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3937 @item @code{relation_equal}
3938 @tab @dots{}a @code{==} relation
3939 @item @code{relation_not_equal}
3940 @tab @dots{}a @code{!=} relation
3941 @item @code{relation_less}
3942 @tab @dots{}a @code{<} relation
3943 @item @code{relation_less_or_equal}
3944 @tab @dots{}a @code{<=} relation
3945 @item @code{relation_greater}
3946 @tab @dots{}a @code{>} relation
3947 @item @code{relation_greater_or_equal}
3948 @tab @dots{}a @code{>=} relation
3950 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3952 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3953 @item @code{polynomial}
3954 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3955 @item @code{integer_polynomial}
3956 @tab @dots{}a polynomial with (non-complex) integer coefficients
3957 @item @code{cinteger_polynomial}
3958 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3959 @item @code{rational_polynomial}
3960 @tab @dots{}a polynomial with (non-complex) rational coefficients
3961 @item @code{crational_polynomial}
3962 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3963 @item @code{rational_function}
3964 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3968 To determine whether an expression is commutative or non-commutative and if
3969 so, with which other expressions it would commutate, you use the methods
3970 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3971 for an explanation of these.
3974 @subsection Accessing subexpressions
3977 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3978 @code{function}, act as containers for subexpressions. For example, the
3979 subexpressions of a sum (an @code{add} object) are the individual terms,
3980 and the subexpressions of a @code{function} are the function's arguments.
3982 @cindex @code{nops()}
3984 GiNaC provides several ways of accessing subexpressions. The first way is to
3989 ex ex::op(size_t i);
3992 @code{nops()} determines the number of subexpressions (operands) contained
3993 in the expression, while @code{op(i)} returns the @code{i}-th
3994 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3995 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3996 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3997 @math{i>0} are the indices.
4000 @cindex @code{const_iterator}
4001 The second way to access subexpressions is via the STL-style random-access
4002 iterator class @code{const_iterator} and the methods
4005 const_iterator ex::begin();
4006 const_iterator ex::end();
4009 @code{begin()} returns an iterator referring to the first subexpression;
4010 @code{end()} returns an iterator which is one-past the last subexpression.
4011 If the expression has no subexpressions, then @code{begin() == end()}. These
4012 iterators can also be used in conjunction with non-modifying STL algorithms.
4014 Here is an example that (non-recursively) prints the subexpressions of a
4015 given expression in three different ways:
4022 for (size_t i = 0; i != e.nops(); ++i)
4023 cout << e.op(i) << endl;
4026 for (const_iterator i = e.begin(); i != e.end(); ++i)
4029 // with iterators and STL copy()
4030 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4034 @cindex @code{const_preorder_iterator}
4035 @cindex @code{const_postorder_iterator}
4036 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4037 expression's immediate children. GiNaC provides two additional iterator
4038 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4039 that iterate over all objects in an expression tree, in preorder or postorder,
4040 respectively. They are STL-style forward iterators, and are created with the
4044 const_preorder_iterator ex::preorder_begin();
4045 const_preorder_iterator ex::preorder_end();
4046 const_postorder_iterator ex::postorder_begin();
4047 const_postorder_iterator ex::postorder_end();
4050 The following example illustrates the differences between
4051 @code{const_iterator}, @code{const_preorder_iterator}, and
4052 @code{const_postorder_iterator}:
4056 symbol A("A"), B("B"), C("C");
4057 ex e = lst@{lst@{A, B@}, C@};
4059 std::copy(e.begin(), e.end(),
4060 std::ostream_iterator<ex>(cout, "\n"));
4064 std::copy(e.preorder_begin(), e.preorder_end(),
4065 std::ostream_iterator<ex>(cout, "\n"));
4072 std::copy(e.postorder_begin(), e.postorder_end(),
4073 std::ostream_iterator<ex>(cout, "\n"));
4082 @cindex @code{relational} (class)
4083 Finally, the left-hand side and right-hand side expressions of objects of
4084 class @code{relational} (and only of these) can also be accessed with the
4093 @subsection Comparing expressions
4094 @cindex @code{is_equal()}
4095 @cindex @code{is_zero()}
4097 Expressions can be compared with the usual C++ relational operators like
4098 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4099 the result is usually not determinable and the result will be @code{false},
4100 except in the case of the @code{!=} operator. You should also be aware that
4101 GiNaC will only do the most trivial test for equality (subtracting both
4102 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4105 Actually, if you construct an expression like @code{a == b}, this will be
4106 represented by an object of the @code{relational} class (@pxref{Relations})
4107 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4109 There are also two methods
4112 bool ex::is_equal(const ex & other);
4116 for checking whether one expression is equal to another, or equal to zero,
4117 respectively. See also the method @code{ex::is_zero_matrix()},
4121 @subsection Ordering expressions
4122 @cindex @code{ex_is_less} (class)
4123 @cindex @code{ex_is_equal} (class)
4124 @cindex @code{compare()}
4126 Sometimes it is necessary to establish a mathematically well-defined ordering
4127 on a set of arbitrary expressions, for example to use expressions as keys
4128 in a @code{std::map<>} container, or to bring a vector of expressions into
4129 a canonical order (which is done internally by GiNaC for sums and products).
4131 The operators @code{<}, @code{>} etc. described in the last section cannot
4132 be used for this, as they don't implement an ordering relation in the
4133 mathematical sense. In particular, they are not guaranteed to be
4134 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4135 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4138 By default, STL classes and algorithms use the @code{<} and @code{==}
4139 operators to compare objects, which are unsuitable for expressions, but GiNaC
4140 provides two functors that can be supplied as proper binary comparison
4141 predicates to the STL:
4146 bool operator()(const ex &lh, const ex &rh) const;
4149 class ex_is_equal @{
4151 bool operator()(const ex &lh, const ex &rh) const;
4155 For example, to define a @code{map} that maps expressions to strings you
4159 std::map<ex, std::string, ex_is_less> myMap;
4162 Omitting the @code{ex_is_less} template parameter will introduce spurious
4163 bugs because the map operates improperly.
4165 Other examples for the use of the functors:
4173 std::sort(v.begin(), v.end(), ex_is_less());
4175 // count the number of expressions equal to '1'
4176 unsigned num_ones = std::count_if(v.begin(), v.end(),
4177 [](const ex& e) @{ return ex_is_equal()(e, 1); @});
4180 The implementation of @code{ex_is_less} uses the member function
4183 int ex::compare(const ex & other) const;
4186 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4187 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4191 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4192 @c node-name, next, previous, up
4193 @section Numerical evaluation
4194 @cindex @code{evalf()}
4196 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4197 To evaluate them using floating-point arithmetic you need to call
4200 ex ex::evalf() const;
4203 @cindex @code{Digits}
4204 The accuracy of the evaluation is controlled by the global object @code{Digits}
4205 which can be assigned an integer value. The default value of @code{Digits}
4206 is 17. @xref{Numbers}, for more information and examples.
4208 To evaluate an expression to a @code{double} floating-point number you can
4209 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4213 // Approximate sin(x/Pi)
4215 ex e = series(sin(x/Pi), x == 0, 6);
4217 // Evaluate numerically at x=0.1
4218 ex f = evalf(e.subs(x == 0.1));
4220 // ex_to<numeric> is an unsafe cast, so check the type first
4221 if (is_a<numeric>(f)) @{
4222 double d = ex_to<numeric>(f).to_double();
4231 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4232 @c node-name, next, previous, up
4233 @section Substituting expressions
4234 @cindex @code{subs()}
4236 Algebraic objects inside expressions can be replaced with arbitrary
4237 expressions via the @code{.subs()} method:
4240 ex ex::subs(const ex & e, unsigned options = 0);
4241 ex ex::subs(const exmap & m, unsigned options = 0);
4242 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4245 In the first form, @code{subs()} accepts a relational of the form
4246 @samp{object == expression} or a @code{lst} of such relationals:
4250 symbol x("x"), y("y");
4252 ex e1 = 2*x*x-4*x+3;
4253 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4257 cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
4262 If you specify multiple substitutions, they are performed in parallel, so e.g.
4263 @code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
4265 The second form of @code{subs()} takes an @code{exmap} object which is a
4266 pair associative container that maps expressions to expressions (currently
4267 implemented as a @code{std::map}). This is the most efficient one of the
4268 three @code{subs()} forms and should be used when the number of objects to
4269 be substituted is large or unknown.
4271 Using this form, the second example from above would look like this:
4275 symbol x("x"), y("y");
4281 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4285 The third form of @code{subs()} takes two lists, one for the objects to be
4286 replaced and one for the expressions to be substituted (both lists must
4287 contain the same number of elements). Using this form, you would write
4291 symbol x("x"), y("y");
4294 cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
4298 The optional last argument to @code{subs()} is a combination of
4299 @code{subs_options} flags. There are three options available:
4300 @code{subs_options::no_pattern} disables pattern matching, which makes
4301 large @code{subs()} operations significantly faster if you are not using
4302 patterns. The second option, @code{subs_options::algebraic} enables
4303 algebraic substitutions in products and powers.
4304 @xref{Pattern matching and advanced substitutions}, for more information
4305 about patterns and algebraic substitutions. The third option,
4306 @code{subs_options::no_index_renaming} disables the feature that dummy
4307 indices are renamed if the substitution could give a result in which a
4308 dummy index occurs more than two times. This is sometimes necessary if
4309 you want to use @code{subs()} to rename your dummy indices.
4311 @code{subs()} performs syntactic substitution of any complete algebraic
4312 object; it does not try to match sub-expressions as is demonstrated by the
4317 symbol x("x"), y("y"), z("z");
4319 ex e1 = pow(x+y, 2);
4320 cout << e1.subs(x+y == 4) << endl;
4323 ex e2 = sin(x)*sin(y)*cos(x);
4324 cout << e2.subs(sin(x) == cos(x)) << endl;
4325 // -> cos(x)^2*sin(y)
4328 cout << e3.subs(x+y == 4) << endl;
4330 // (and not 4+z as one might expect)
4334 A more powerful form of substitution using wildcards is described in the
4338 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4339 @c node-name, next, previous, up
4340 @section Pattern matching and advanced substitutions
4341 @cindex @code{wildcard} (class)
4342 @cindex Pattern matching
4344 GiNaC allows the use of patterns for checking whether an expression is of a
4345 certain form or contains subexpressions of a certain form, and for
4346 substituting expressions in a more general way.
4348 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4349 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4350 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4351 an unsigned integer number to allow having multiple different wildcards in a
4352 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4353 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4357 ex wild(unsigned label = 0);
4360 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4363 Some examples for patterns:
4365 @multitable @columnfractions .5 .5
4366 @item @strong{Constructed as} @tab @strong{Output as}
4367 @item @code{wild()} @tab @samp{$0}
4368 @item @code{pow(x,wild())} @tab @samp{x^$0}
4369 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4370 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4376 @item Wildcards behave like symbols and are subject to the same algebraic
4377 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4378 @item As shown in the last example, to use wildcards for indices you have to
4379 use them as the value of an @code{idx} object. This is because indices must
4380 always be of class @code{idx} (or a subclass).
4381 @item Wildcards only represent expressions or subexpressions. It is not
4382 possible to use them as placeholders for other properties like index
4383 dimension or variance, representation labels, symmetry of indexed objects
4385 @item Because wildcards are commutative, it is not possible to use wildcards
4386 as part of noncommutative products.
4387 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4388 are also valid patterns.
4391 @subsection Matching expressions
4392 @cindex @code{match()}
4393 The most basic application of patterns is to check whether an expression
4394 matches a given pattern. This is done by the function
4397 bool ex::match(const ex & pattern);
4398 bool ex::match(const ex & pattern, exmap& repls);
4401 This function returns @code{true} when the expression matches the pattern
4402 and @code{false} if it doesn't. If used in the second form, the actual
4403 subexpressions matched by the wildcards get returned in the associative
4404 array @code{repls} with @samp{wildcard} as a key. If @code{match()}
4405 returns false, @code{repls} remains unmodified.
4407 The matching algorithm works as follows:
4410 @item A single wildcard matches any expression. If one wildcard appears
4411 multiple times in a pattern, it must match the same expression in all
4412 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4413 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4414 @item If the expression is not of the same class as the pattern, the match
4415 fails (i.e. a sum only matches a sum, a function only matches a function,
4417 @item If the pattern is a function, it only matches the same function
4418 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4419 @item Except for sums and products, the match fails if the number of
4420 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4422 @item If there are no subexpressions, the expressions and the pattern must
4423 be equal (in the sense of @code{is_equal()}).
4424 @item Except for sums and products, each subexpression (@code{op()}) must
4425 match the corresponding subexpression of the pattern.
4428 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4429 account for their commutativity and associativity:
4432 @item If the pattern contains a term or factor that is a single wildcard,
4433 this one is used as the @dfn{global wildcard}. If there is more than one
4434 such wildcard, one of them is chosen as the global wildcard in a random
4436 @item Every term/factor of the pattern, except the global wildcard, is
4437 matched against every term of the expression in sequence. If no match is
4438 found, the whole match fails. Terms that did match are not considered in
4440 @item If there are no unmatched terms left, the match succeeds. Otherwise
4441 the match fails unless there is a global wildcard in the pattern, in
4442 which case this wildcard matches the remaining terms.
4445 In general, having more than one single wildcard as a term of a sum or a
4446 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4449 Here are some examples in @command{ginsh} to demonstrate how it works (the
4450 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4451 match fails, and the list of wildcard replacements otherwise):
4454 > match((x+y)^a,(x+y)^a);
4456 > match((x+y)^a,(x+y)^b);
4458 > match((x+y)^a,$1^$2);
4460 > match((x+y)^a,$1^$1);
4462 > match((x+y)^(x+y),$1^$1);
4464 > match((x+y)^(x+y),$1^$2);
4466 > match((a+b)*(a+c),($1+b)*($1+c));
4468 > match((a+b)*(a+c),(a+$1)*(a+$2));
4470 (Unpredictable. The result might also be [$1==c,$2==b].)
4471 > match((a+b)*(a+c),($1+$2)*($1+$3));
4472 (The result is undefined. Due to the sequential nature of the algorithm
4473 and the re-ordering of terms in GiNaC, the match for the first factor
4474 may be @{$1==a,$2==b@} in which case the match for the second factor
4475 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4477 > match(a*(x+y)+a*z+b,a*$1+$2);
4478 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4479 @{$1=x+y,$2=a*z+b@}.)
4480 > match(a+b+c+d+e+f,c);
4482 > match(a+b+c+d+e+f,c+$0);
4484 > match(a+b+c+d+e+f,c+e+$0);
4486 > match(a+b,a+b+$0);
4488 > match(a*b^2,a^$1*b^$2);
4490 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4491 even though a==a^1.)
4492 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4494 > match(atan2(y,x^2),atan2(y,$0));
4498 @subsection Matching parts of expressions
4499 @cindex @code{has()}
4500 A more general way to look for patterns in expressions is provided by the
4504 bool ex::has(const ex & pattern);
4507 This function checks whether a pattern is matched by an expression itself or
4508 by any of its subexpressions.
4510 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4511 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4514 > has(x*sin(x+y+2*a),y);
4516 > has(x*sin(x+y+2*a),x+y);
4518 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4519 has the subexpressions "x", "y" and "2*a".)
4520 > has(x*sin(x+y+2*a),x+y+$1);
4522 (But this is possible.)
4523 > has(x*sin(2*(x+y)+2*a),x+y);
4525 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4526 which "x+y" is not a subexpression.)
4529 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4531 > has(4*x^2-x+3,$1*x);
4533 > has(4*x^2+x+3,$1*x);
4535 (Another possible pitfall. The first expression matches because the term
4536 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4537 contains a linear term you should use the coeff() function instead.)
4540 @cindex @code{find()}
4544 bool ex::find(const ex & pattern, exset& found);
4547 works a bit like @code{has()} but it doesn't stop upon finding the first
4548 match. Instead, it appends all found matches to the specified list. If there
4549 are multiple occurrences of the same expression, it is entered only once to
4550 the list. @code{find()} returns false if no matches were found (in
4551 @command{ginsh}, it returns an empty list):
4554 > find(1+x+x^2+x^3,x);
4556 > find(1+x+x^2+x^3,y);
4558 > find(1+x+x^2+x^3,x^$1);
4560 (Note the absence of "x".)
4561 > expand((sin(x)+sin(y))*(a+b));
4562 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4567 @subsection Substituting expressions
4568 @cindex @code{subs()}
4569 Probably the most useful application of patterns is to use them for
4570 substituting expressions with the @code{subs()} method. Wildcards can be
4571 used in the search patterns as well as in the replacement expressions, where
4572 they get replaced by the expressions matched by them. @code{subs()} doesn't
4573 know anything about algebra; it performs purely syntactic substitutions.
4578 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4580 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4582 > subs((a+b+c)^2,a+b==x);
4584 > subs((a+b+c)^2,a+b+$1==x+$1);
4586 > subs(a+2*b,a+b==x);
4588 > subs(4*x^3-2*x^2+5*x-1,x==a);
4590 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4592 > subs(sin(1+sin(x)),sin($1)==cos($1));
4594 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4598 The last example would be written in C++ in this way:
4602 symbol a("a"), b("b"), x("x"), y("y");
4603 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4604 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4605 cout << e.expand() << endl;
4610 @subsection The option algebraic
4611 Both @code{has()} and @code{subs()} take an optional argument to pass them
4612 extra options. This section describes what happens if you give the former
4613 the option @code{has_options::algebraic} or the latter
4614 @code{subs_options::algebraic}. In that case the matching condition for
4615 powers and multiplications is changed in such a way that they become
4616 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4617 If you use these options you will find that
4618 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4619 Besides matching some of the factors of a product also powers match as
4620 often as is possible without getting negative exponents. For example
4621 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4622 @code{x*c^2*z}. This also works with negative powers:
4623 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4624 return @code{x^(-1)*c^2*z}.
4626 @strong{Please notice:} this only works for multiplications
4627 and not for locating @code{x+y} within @code{x+y+z}.
4630 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4631 @c node-name, next, previous, up
4632 @section Applying a function on subexpressions
4633 @cindex tree traversal
4634 @cindex @code{map()}
4636 Sometimes you may want to perform an operation on specific parts of an
4637 expression while leaving the general structure of it intact. An example
4638 of this would be a matrix trace operation: the trace of a sum is the sum
4639 of the traces of the individual terms. That is, the trace should @dfn{map}
4640 on the sum, by applying itself to each of the sum's operands. It is possible
4641 to do this manually which usually results in code like this:
4646 if (is_a<matrix>(e))
4647 return ex_to<matrix>(e).trace();
4648 else if (is_a<add>(e)) @{
4650 for (size_t i=0; i<e.nops(); i++)
4651 sum += calc_trace(e.op(i));
4653 @} else if (is_a<mul>)(e)) @{
4661 This is, however, slightly inefficient (if the sum is very large it can take
4662 a long time to add the terms one-by-one), and its applicability is limited to
4663 a rather small class of expressions. If @code{calc_trace()} is called with
4664 a relation or a list as its argument, you will probably want the trace to
4665 be taken on both sides of the relation or of all elements of the list.
4667 GiNaC offers the @code{map()} method to aid in the implementation of such
4671 ex ex::map(map_function & f) const;
4672 ex ex::map(ex (*f)(const ex & e)) const;
4675 In the first (preferred) form, @code{map()} takes a function object that
4676 is subclassed from the @code{map_function} class. In the second form, it
4677 takes a pointer to a function that accepts and returns an expression.
4678 @code{map()} constructs a new expression of the same type, applying the
4679 specified function on all subexpressions (in the sense of @code{op()}),
4682 The use of a function object makes it possible to supply more arguments to
4683 the function that is being mapped, or to keep local state information.
4684 The @code{map_function} class declares a virtual function call operator
4685 that you can overload. Here is a sample implementation of @code{calc_trace()}
4686 that uses @code{map()} in a recursive fashion:
4689 struct calc_trace : public map_function @{
4690 ex operator()(const ex &e)
4692 if (is_a<matrix>(e))
4693 return ex_to<matrix>(e).trace();
4694 else if (is_a<mul>(e)) @{
4697 return e.map(*this);
4702 This function object could then be used like this:
4706 ex M = ... // expression with matrices
4707 calc_trace do_trace;
4708 ex tr = do_trace(M);
4712 Here is another example for you to meditate over. It removes quadratic
4713 terms in a variable from an expanded polynomial:
4716 struct map_rem_quad : public map_function @{
4718 map_rem_quad(const ex & var_) : var(var_) @{@}
4720 ex operator()(const ex & e)
4722 if (is_a<add>(e) || is_a<mul>(e))
4723 return e.map(*this);
4724 else if (is_a<power>(e) &&
4725 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4735 symbol x("x"), y("y");
4738 for (int i=0; i<8; i++)
4739 e += pow(x, i) * pow(y, 8-i) * (i+1);
4741 // -> 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
4743 map_rem_quad rem_quad(x);
4744 cout << rem_quad(e) << endl;
4745 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4749 @command{ginsh} offers a slightly different implementation of @code{map()}
4750 that allows applying algebraic functions to operands. The second argument
4751 to @code{map()} is an expression containing the wildcard @samp{$0} which
4752 acts as the placeholder for the operands:
4757 > map(a+2*b,sin($0));
4759 > map(@{a,b,c@},$0^2+$0);
4760 @{a^2+a,b^2+b,c^2+c@}
4763 Note that it is only possible to use algebraic functions in the second
4764 argument. You can not use functions like @samp{diff()}, @samp{op()},
4765 @samp{subs()} etc. because these are evaluated immediately:
4768 > map(@{a,b,c@},diff($0,a));
4770 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4771 to "map(@{a,b,c@},0)".
4775 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4776 @c node-name, next, previous, up
4777 @section Visitors and tree traversal
4778 @cindex tree traversal
4779 @cindex @code{visitor} (class)
4780 @cindex @code{accept()}
4781 @cindex @code{visit()}
4782 @cindex @code{traverse()}
4783 @cindex @code{traverse_preorder()}
4784 @cindex @code{traverse_postorder()}
4786 Suppose that you need a function that returns a list of all indices appearing
4787 in an arbitrary expression. The indices can have any dimension, and for
4788 indices with variance you always want the covariant version returned.
4790 You can't use @code{get_free_indices()} because you also want to include
4791 dummy indices in the list, and you can't use @code{find()} as it needs
4792 specific index dimensions (and it would require two passes: one for indices
4793 with variance, one for plain ones).
4795 The obvious solution to this problem is a tree traversal with a type switch,
4796 such as the following:
4799 void gather_indices_helper(const ex & e, lst & l)
4801 if (is_a<varidx>(e)) @{
4802 const varidx & vi = ex_to<varidx>(e);
4803 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4804 @} else if (is_a<idx>(e)) @{
4807 size_t n = e.nops();
4808 for (size_t i = 0; i < n; ++i)
4809 gather_indices_helper(e.op(i), l);
4813 lst gather_indices(const ex & e)
4816 gather_indices_helper(e, l);
4823 This works fine but fans of object-oriented programming will feel
4824 uncomfortable with the type switch. One reason is that there is a possibility
4825 for subtle bugs regarding derived classes. If we had, for example, written
4828 if (is_a<idx>(e)) @{
4830 @} else if (is_a<varidx>(e)) @{
4834 in @code{gather_indices_helper}, the code wouldn't have worked because the
4835 first line "absorbs" all classes derived from @code{idx}, including
4836 @code{varidx}, so the special case for @code{varidx} would never have been
4839 Also, for a large number of classes, a type switch like the above can get
4840 unwieldy and inefficient (it's a linear search, after all).
4841 @code{gather_indices_helper} only checks for two classes, but if you had to
4842 write a function that required a different implementation for nearly
4843 every GiNaC class, the result would be very hard to maintain and extend.
4845 The cleanest approach to the problem would be to add a new virtual function
4846 to GiNaC's class hierarchy. In our example, there would be specializations
4847 for @code{idx} and @code{varidx} while the default implementation in
4848 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4849 impossible to add virtual member functions to existing classes without
4850 changing their source and recompiling everything. GiNaC comes with source,
4851 so you could actually do this, but for a small algorithm like the one
4852 presented this would be impractical.
4854 One solution to this dilemma is the @dfn{Visitor} design pattern,
4855 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4856 variation, described in detail in
4857 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4858 virtual functions to the class hierarchy to implement operations, GiNaC
4859 provides a single "bouncing" method @code{accept()} that takes an instance
4860 of a special @code{visitor} class and redirects execution to the one
4861 @code{visit()} virtual function of the visitor that matches the type of
4862 object that @code{accept()} was being invoked on.
4864 Visitors in GiNaC must derive from the global @code{visitor} class as well
4865 as from the class @code{T::visitor} of each class @code{T} they want to
4866 visit, and implement the member functions @code{void visit(const T &)} for
4872 void ex::accept(visitor & v) const;
4875 will then dispatch to the correct @code{visit()} member function of the
4876 specified visitor @code{v} for the type of GiNaC object at the root of the
4877 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4879 Here is an example of a visitor:
4883 : public visitor, // this is required
4884 public add::visitor, // visit add objects
4885 public numeric::visitor, // visit numeric objects
4886 public basic::visitor // visit basic objects
4888 void visit(const add & x)
4889 @{ cout << "called with an add object" << endl; @}
4891 void visit(const numeric & x)
4892 @{ cout << "called with a numeric object" << endl; @}
4894 void visit(const basic & x)
4895 @{ cout << "called with a basic object" << endl; @}
4899 which can be used as follows:
4910 // prints "called with a numeric object"
4912 // prints "called with an add object"
4914 // prints "called with a basic object"
4918 The @code{visit(const basic &)} method gets called for all objects that are
4919 not @code{numeric} or @code{add} and acts as an (optional) default.
4921 From a conceptual point of view, the @code{visit()} methods of the visitor
4922 behave like a newly added virtual function of the visited hierarchy.
4923 In addition, visitors can store state in member variables, and they can
4924 be extended by deriving a new visitor from an existing one, thus building
4925 hierarchies of visitors.
4927 We can now rewrite our index example from above with a visitor:
4930 class gather_indices_visitor
4931 : public visitor, public idx::visitor, public varidx::visitor
4935 void visit(const idx & i)
4940 void visit(const varidx & vi)
4942 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4946 const lst & get_result() // utility function
4955 What's missing is the tree traversal. We could implement it in
4956 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4959 void ex::traverse_preorder(visitor & v) const;
4960 void ex::traverse_postorder(visitor & v) const;
4961 void ex::traverse(visitor & v) const;
4964 @code{traverse_preorder()} visits a node @emph{before} visiting its
4965 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4966 visiting its subexpressions. @code{traverse()} is a synonym for
4967 @code{traverse_preorder()}.
4969 Here is a new implementation of @code{gather_indices()} that uses the visitor
4970 and @code{traverse()}:
4973 lst gather_indices(const ex & e)
4975 gather_indices_visitor v;
4977 return v.get_result();
4981 Alternatively, you could use pre- or postorder iterators for the tree
4985 lst gather_indices(const ex & e)
4987 gather_indices_visitor v;
4988 for (const_preorder_iterator i = e.preorder_begin();
4989 i != e.preorder_end(); ++i) @{
4992 return v.get_result();
4997 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4998 @c node-name, next, previous, up
4999 @section Polynomial arithmetic
5001 @subsection Testing whether an expression is a polynomial
5002 @cindex @code{is_polynomial()}
5004 Testing whether an expression is a polynomial in one or more variables
5005 can be done with the method
5007 bool ex::is_polynomial(const ex & vars) const;
5009 In the case of more than
5010 one variable, the variables are given as a list.
5013 (x*y*sin(y)).is_polynomial(x) // Returns true.
5014 (x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
5017 @subsection Expanding and collecting
5018 @cindex @code{expand()}
5019 @cindex @code{collect()}
5020 @cindex @code{collect_common_factors()}
5022 A polynomial in one or more variables has many equivalent
5023 representations. Some useful ones serve a specific purpose. Consider
5024 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5025 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5026 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5027 representations are the recursive ones where one collects for exponents
5028 in one of the three variable. Since the factors are themselves
5029 polynomials in the remaining two variables the procedure can be
5030 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5031 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5034 To bring an expression into expanded form, its method
5037 ex ex::expand(unsigned options = 0);
5040 may be called. In our example above, this corresponds to @math{4*x*y +
5041 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5042 GiNaC is not easy to guess you should be prepared to see different
5043 orderings of terms in such sums!
5045 Another useful representation of multivariate polynomials is as a
5046 univariate polynomial in one of the variables with the coefficients
5047 being polynomials in the remaining variables. The method
5048 @code{collect()} accomplishes this task:
5051 ex ex::collect(const ex & s, bool distributed = false);
5054 The first argument to @code{collect()} can also be a list of objects in which
5055 case the result is either a recursively collected polynomial, or a polynomial
5056 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5057 by the @code{distributed} flag.
5059 Note that the original polynomial needs to be in expanded form (for the
5060 variables concerned) in order for @code{collect()} to be able to find the
5061 coefficients properly.
5063 The following @command{ginsh} transcript shows an application of @code{collect()}
5064 together with @code{find()}:
5067 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5068 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)
5069 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5070 > collect(a,@{p,q@});
5071 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5072 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5073 > collect(a,find(a,sin($1)));
5074 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5075 > collect(a,@{find(a,sin($1)),p,q@});
5076 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5077 > collect(a,@{find(a,sin($1)),d@});
5078 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5081 Polynomials can often be brought into a more compact form by collecting
5082 common factors from the terms of sums. This is accomplished by the function
5085 ex collect_common_factors(const ex & e);
5088 This function doesn't perform a full factorization but only looks for
5089 factors which are already explicitly present:
5092 > collect_common_factors(a*x+a*y);
5094 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5096 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5097 (c+a)*a*(x*y+y^2+x)*b
5100 @subsection Degree and coefficients
5101 @cindex @code{degree()}
5102 @cindex @code{ldegree()}
5103 @cindex @code{coeff()}
5105 The degree and low degree of a polynomial in expanded form can be obtained
5106 using the two methods
5109 int ex::degree(const ex & s);
5110 int ex::ldegree(const ex & s);
5113 These functions even work on rational functions, returning the asymptotic
5114 degree. By definition, the degree of zero is zero. To extract a coefficient
5115 with a certain power from an expanded polynomial you use
5118 ex ex::coeff(const ex & s, int n);
5121 You can also obtain the leading and trailing coefficients with the methods
5124 ex ex::lcoeff(const ex & s);
5125 ex ex::tcoeff(const ex & s);
5128 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5131 An application is illustrated in the next example, where a multivariate
5132 polynomial is analyzed:
5136 symbol x("x"), y("y");
5137 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5138 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5139 ex Poly = PolyInp.expand();
5141 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5142 cout << "The x^" << i << "-coefficient is "
5143 << Poly.coeff(x,i) << endl;
5145 cout << "As polynomial in y: "
5146 << Poly.collect(y) << endl;
5150 When run, it returns an output in the following fashion:
5153 The x^0-coefficient is y^2+11*y
5154 The x^1-coefficient is 5*y^2-2*y
5155 The x^2-coefficient is -1
5156 The x^3-coefficient is 4*y
5157 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5160 As always, the exact output may vary between different versions of GiNaC
5161 or even from run to run since the internal canonical ordering is not
5162 within the user's sphere of influence.
5164 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5165 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5166 with non-polynomial expressions as they not only work with symbols but with
5167 constants, functions and indexed objects as well:
5171 symbol a("a"), b("b"), c("c"), x("x");
5172 idx i(symbol("i"), 3);
5174 ex e = pow(sin(x) - cos(x), 4);
5175 cout << e.degree(cos(x)) << endl;
5177 cout << e.expand().coeff(sin(x), 3) << endl;
5180 e = indexed(a+b, i) * indexed(b+c, i);
5181 e = e.expand(expand_options::expand_indexed);
5182 cout << e.collect(indexed(b, i)) << endl;
5183 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5188 @subsection Polynomial division
5189 @cindex polynomial division
5192 @cindex pseudo-remainder
5193 @cindex @code{quo()}
5194 @cindex @code{rem()}
5195 @cindex @code{prem()}
5196 @cindex @code{divide()}
5201 ex quo(const ex & a, const ex & b, const ex & x);
5202 ex rem(const ex & a, const ex & b, const ex & x);
5205 compute the quotient and remainder of univariate polynomials in the variable
5206 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5208 The additional function
5211 ex prem(const ex & a, const ex & b, const ex & x);
5214 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5215 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5217 Exact division of multivariate polynomials is performed by the function
5220 bool divide(const ex & a, const ex & b, ex & q);
5223 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5224 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5225 in which case the value of @code{q} is undefined.
5228 @subsection Unit, content and primitive part
5229 @cindex @code{unit()}
5230 @cindex @code{content()}
5231 @cindex @code{primpart()}
5232 @cindex @code{unitcontprim()}
5237 ex ex::unit(const ex & x);
5238 ex ex::content(const ex & x);
5239 ex ex::primpart(const ex & x);
5240 ex ex::primpart(const ex & x, const ex & c);
5243 return the unit part, content part, and primitive polynomial of a multivariate
5244 polynomial with respect to the variable @samp{x} (the unit part being the sign
5245 of the leading coefficient, the content part being the GCD of the coefficients,
5246 and the primitive polynomial being the input polynomial divided by the unit and
5247 content parts). The second variant of @code{primpart()} expects the previously
5248 calculated content part of the polynomial in @code{c}, which enables it to
5249 work faster in the case where the content part has already been computed. The
5250 product of unit, content, and primitive part is the original polynomial.
5252 Additionally, the method
5255 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5258 computes the unit, content, and primitive parts in one go, returning them
5259 in @code{u}, @code{c}, and @code{p}, respectively.
5262 @subsection GCD, LCM and resultant
5265 @cindex @code{gcd()}
5266 @cindex @code{lcm()}
5268 The functions for polynomial greatest common divisor and least common
5269 multiple have the synopsis
5272 ex gcd(const ex & a, const ex & b);
5273 ex lcm(const ex & a, const ex & b);
5276 The functions @code{gcd()} and @code{lcm()} accept two expressions
5277 @code{a} and @code{b} as arguments and return a new expression, their
5278 greatest common divisor or least common multiple, respectively. If the
5279 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5280 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5281 the coefficients must be rationals.
5284 #include <ginac/ginac.h>
5285 using namespace GiNaC;
5289 symbol x("x"), y("y"), z("z");
5290 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5291 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5293 ex P_gcd = gcd(P_a, P_b);
5295 ex P_lcm = lcm(P_a, P_b);
5296 // 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
5301 @cindex @code{resultant()}
5303 The resultant of two expressions only makes sense with polynomials.
5304 It is always computed with respect to a specific symbol within the
5305 expressions. The function has the interface
5308 ex resultant(const ex & a, const ex & b, const ex & s);
5311 Resultants are symmetric in @code{a} and @code{b}. The following example
5312 computes the resultant of two expressions with respect to @code{x} and
5313 @code{y}, respectively:
5316 #include <ginac/ginac.h>
5317 using namespace GiNaC;
5321 symbol x("x"), y("y");
5323 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5326 r = resultant(e1, e2, x);
5328 r = resultant(e1, e2, y);
5333 @subsection Square-free decomposition
5334 @cindex square-free decomposition
5335 @cindex factorization
5336 @cindex @code{sqrfree()}
5338 Square-free decomposition is available in GiNaC:
5340 ex sqrfree(const ex & a, const lst & l = lst@{@});
5342 Here is an example that by the way illustrates how the exact form of the
5343 result may slightly depend on the order of differentiation, calling for
5344 some care with subsequent processing of the result:
5347 symbol x("x"), y("y");
5348 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5350 cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
5351 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5353 cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
5354 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5356 cout << sqrfree(BiVarPol) << endl;
5357 // -> depending on luck, any of the above
5360 Note also, how factors with the same exponents are not fully factorized
5363 @subsection Polynomial factorization
5364 @cindex factorization
5365 @cindex polynomial factorization
5366 @cindex @code{factor()}
5368 Polynomials can also be fully factored with a call to the function
5370 ex factor(const ex & a, unsigned int options = 0);
5372 The factorization works for univariate and multivariate polynomials with
5373 rational coefficients. The following code snippet shows its capabilities:
5376 cout << factor(pow(x,2)-1) << endl;
5378 cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
5379 // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
5380 cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
5381 // -> -1+sin(-1+x^2)+x^2
5384 The results are as expected except for the last one where no factorization
5385 seems to have been done. This is due to the default option
5386 @command{factor_options::polynomial} (equals zero) to @command{factor()}, which
5387 tells GiNaC to try a factorization only if the expression is a valid polynomial.
5388 In the shown example this is not the case, because one term is a function.
5390 There exists a second option @command{factor_options::all}, which tells GiNaC to
5391 ignore non-polynomial parts of an expression and also to look inside function
5392 arguments. With this option the example gives:
5395 cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
5397 // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
5400 GiNaC's factorization functions cannot handle algebraic extensions. Therefore
5401 the following example does not factor:
5404 cout << factor(pow(x,2)-2) << endl;
5405 // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
5408 Factorization is useful in many applications. A lot of algorithms in computer
5409 algebra depend on the ability to factor a polynomial. Of course, factorization
5410 can also be used to simplify expressions, but it is costly and applying it to
5411 complicated expressions (high degrees or many terms) may consume far too much
5412 time. So usually, looking for a GCD at strategic points in a calculation is the
5413 cheaper and more appropriate alternative.
5415 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5416 @c node-name, next, previous, up
5417 @section Rational expressions
5419 @subsection The @code{normal} method
5420 @cindex @code{normal()}
5421 @cindex simplification
5422 @cindex temporary replacement
5424 Some basic form of simplification of expressions is called for frequently.
5425 GiNaC provides the method @code{.normal()}, which converts a rational function
5426 into an equivalent rational function of the form @samp{numerator/denominator}
5427 where numerator and denominator are coprime. If the input expression is already
5428 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5429 otherwise it performs fraction addition and multiplication.
5431 @code{.normal()} can also be used on expressions which are not rational functions
5432 as it will replace all non-rational objects (like functions or non-integer
5433 powers) by temporary symbols to bring the expression to the domain of rational
5434 functions before performing the normalization, and re-substituting these
5435 symbols afterwards. This algorithm is also available as a separate method
5436 @code{.to_rational()}, described below.
5438 This means that both expressions @code{t1} and @code{t2} are indeed
5439 simplified in this little code snippet:
5444 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5445 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5446 std::cout << "t1 is " << t1.normal() << std::endl;
5447 std::cout << "t2 is " << t2.normal() << std::endl;
5451 Of course this works for multivariate polynomials too, so the ratio of
5452 the sample-polynomials from the section about GCD and LCM above would be
5453 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5456 @subsection Numerator and denominator
5459 @cindex @code{numer()}
5460 @cindex @code{denom()}
5461 @cindex @code{numer_denom()}
5463 The numerator and denominator of an expression can be obtained with
5468 ex ex::numer_denom();
5471 These functions will first normalize the expression as described above and
5472 then return the numerator, denominator, or both as a list, respectively.
5473 If you need both numerator and denominator, call @code{numer_denom()}: it
5474 is faster than using @code{numer()} and @code{denom()} separately. And even
5475 more important: a separate evaluation of @code{numer()} and @code{denom()}
5476 may result in a spurious sign, e.g. for $x/(x^2-1)$ @code{numer()} may
5477 return $x$ and @code{denom()} $1-x^2$.
5480 @subsection Converting to a polynomial or rational expression
5481 @cindex @code{to_polynomial()}
5482 @cindex @code{to_rational()}
5484 Some of the methods described so far only work on polynomials or rational
5485 functions. GiNaC provides a way to extend the domain of these functions to
5486 general expressions by using the temporary replacement algorithm described
5487 above. You do this by calling
5490 ex ex::to_polynomial(exmap & m);
5494 ex ex::to_rational(exmap & m);
5497 on the expression to be converted. The supplied @code{exmap} will be filled
5498 with the generated temporary symbols and their replacement expressions in a
5499 format that can be used directly for the @code{subs()} method. It can also
5500 already contain a list of replacements from an earlier application of
5501 @code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
5502 it on multiple expressions and get consistent results.
5504 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5505 is probably best illustrated with an example:
5509 symbol x("x"), y("y");
5510 ex a = 2*x/sin(x) - y/(3*sin(x));
5514 ex p = a.to_polynomial(mp);
5515 cout << " = " << p << "\n with " << mp << endl;
5516 // = symbol3*symbol2*y+2*symbol2*x
5517 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5520 ex r = a.to_rational(mr);
5521 cout << " = " << r << "\n with " << mr << endl;
5522 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5523 // with @{symbol4==sin(x)@}
5527 The following more useful example will print @samp{sin(x)-cos(x)}:
5532 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5533 ex b = sin(x) + cos(x);
5536 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5537 cout << q.subs(m) << endl;
5542 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5543 @c node-name, next, previous, up
5544 @section Symbolic differentiation
5545 @cindex differentiation
5546 @cindex @code{diff()}
5548 @cindex product rule
5550 GiNaC's objects know how to differentiate themselves. Thus, a
5551 polynomial (class @code{add}) knows that its derivative is the sum of
5552 the derivatives of all the monomials:
5556 symbol x("x"), y("y"), z("z");
5557 ex P = pow(x, 5) + pow(x, 2) + y;
5559 cout << P.diff(x,2) << endl;
5561 cout << P.diff(y) << endl; // 1
5563 cout << P.diff(z) << endl; // 0
5568 If a second integer parameter @var{n} is given, the @code{diff} method
5569 returns the @var{n}th derivative.
5571 If @emph{every} object and every function is told what its derivative
5572 is, all derivatives of composed objects can be calculated using the
5573 chain rule and the product rule. Consider, for instance the expression
5574 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5575 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5576 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5577 out that the composition is the generating function for Euler Numbers,
5578 i.e. the so called @var{n}th Euler number is the coefficient of
5579 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5580 identity to code a function that generates Euler numbers in just three
5583 @cindex Euler numbers
5585 #include <ginac/ginac.h>
5586 using namespace GiNaC;
5588 ex EulerNumber(unsigned n)
5591 const ex generator = pow(cosh(x),-1);
5592 return generator.diff(x,n).subs(x==0);
5597 for (unsigned i=0; i<11; i+=2)
5598 std::cout << EulerNumber(i) << std::endl;
5603 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5604 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5605 @code{i} by two since all odd Euler numbers vanish anyways.
5608 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5609 @c node-name, next, previous, up
5610 @section Series expansion
5611 @cindex @code{series()}
5612 @cindex Taylor expansion
5613 @cindex Laurent expansion
5614 @cindex @code{pseries} (class)
5615 @cindex @code{Order()}
5617 Expressions know how to expand themselves as a Taylor series or (more
5618 generally) a Laurent series. As in most conventional Computer Algebra
5619 Systems, no distinction is made between those two. There is a class of
5620 its own for storing such series (@code{class pseries}) and a built-in
5621 function (called @code{Order}) for storing the order term of the series.
5622 As a consequence, if you want to work with series, i.e. multiply two
5623 series, you need to call the method @code{ex::series} again to convert
5624 it to a series object with the usual structure (expansion plus order
5625 term). A sample application from special relativity could read:
5628 #include <ginac/ginac.h>
5629 using namespace std;
5630 using namespace GiNaC;
5634 symbol v("v"), c("c");
5636 ex gamma = 1/sqrt(1 - pow(v/c,2));
5637 ex mass_nonrel = gamma.series(v==0, 10);
5639 cout << "the relativistic mass increase with v is " << endl
5640 << mass_nonrel << endl;
5642 cout << "the inverse square of this series is " << endl
5643 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5647 Only calling the series method makes the last output simplify to
5648 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5649 series raised to the power @math{-2}.
5651 @cindex Machin's formula
5652 As another instructive application, let us calculate the numerical
5653 value of Archimedes' constant
5660 (for which there already exists the built-in constant @code{Pi})
5661 using John Machin's amazing formula
5663 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5666 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5668 This equation (and similar ones) were used for over 200 years for
5669 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5670 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5671 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5672 order term with it and the question arises what the system is supposed
5673 to do when the fractions are plugged into that order term. The solution
5674 is to use the function @code{series_to_poly()} to simply strip the order
5678 #include <ginac/ginac.h>
5679 using namespace GiNaC;
5681 ex machin_pi(int degr)
5684 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5685 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5686 -4*pi_expansion.subs(x==numeric(1,239));
5692 using std::cout; // just for fun, another way of...
5693 using std::endl; // ...dealing with this namespace std.
5695 for (int i=2; i<12; i+=2) @{
5696 pi_frac = machin_pi(i);
5697 cout << i << ":\t" << pi_frac << endl
5698 << "\t" << pi_frac.evalf() << endl;
5704 Note how we just called @code{.series(x,degr)} instead of
5705 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5706 method @code{series()}: if the first argument is a symbol the expression
5707 is expanded in that symbol around point @code{0}. When you run this
5708 program, it will type out:
5712 3.1832635983263598326
5713 4: 5359397032/1706489875
5714 3.1405970293260603143
5715 6: 38279241713339684/12184551018734375
5716 3.141621029325034425
5717 8: 76528487109180192540976/24359780855939418203125
5718 3.141591772182177295
5719 10: 327853873402258685803048818236/104359128170408663038552734375
5720 3.1415926824043995174
5724 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5725 @c node-name, next, previous, up
5726 @section Symmetrization
5727 @cindex @code{symmetrize()}
5728 @cindex @code{antisymmetrize()}
5729 @cindex @code{symmetrize_cyclic()}
5734 ex ex::symmetrize(const lst & l);
5735 ex ex::antisymmetrize(const lst & l);
5736 ex ex::symmetrize_cyclic(const lst & l);
5739 symmetrize an expression by returning the sum over all symmetric,
5740 antisymmetric or cyclic permutations of the specified list of objects,
5741 weighted by the number of permutations.
5743 The three additional methods
5746 ex ex::symmetrize();
5747 ex ex::antisymmetrize();
5748 ex ex::symmetrize_cyclic();
5751 symmetrize or antisymmetrize an expression over its free indices.
5753 Symmetrization is most useful with indexed expressions but can be used with
5754 almost any kind of object (anything that is @code{subs()}able):
5758 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5759 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5761 cout << ex(indexed(A, i, j)).symmetrize() << endl;
5762 // -> 1/2*A.j.i+1/2*A.i.j
5763 cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
5764 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5765 cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
5766 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5772 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5773 @c node-name, next, previous, up
5774 @section Predefined mathematical functions
5776 @subsection Overview
5778 GiNaC contains the following predefined mathematical functions:
5781 @multitable @columnfractions .30 .70
5782 @item @strong{Name} @tab @strong{Function}
5785 @cindex @code{abs()}
5786 @item @code{step(x)}
5788 @cindex @code{step()}
5789 @item @code{csgn(x)}
5791 @cindex @code{conjugate()}
5792 @item @code{conjugate(x)}
5793 @tab complex conjugation
5794 @cindex @code{real_part()}
5795 @item @code{real_part(x)}
5797 @cindex @code{imag_part()}
5798 @item @code{imag_part(x)}
5800 @item @code{sqrt(x)}
5801 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5802 @cindex @code{sqrt()}
5805 @cindex @code{sin()}
5808 @cindex @code{cos()}
5811 @cindex @code{tan()}
5812 @item @code{asin(x)}
5814 @cindex @code{asin()}
5815 @item @code{acos(x)}
5817 @cindex @code{acos()}
5818 @item @code{atan(x)}
5819 @tab inverse tangent
5820 @cindex @code{atan()}
5821 @item @code{atan2(y, x)}
5822 @tab inverse tangent with two arguments
5823 @item @code{sinh(x)}
5824 @tab hyperbolic sine
5825 @cindex @code{sinh()}
5826 @item @code{cosh(x)}
5827 @tab hyperbolic cosine
5828 @cindex @code{cosh()}
5829 @item @code{tanh(x)}
5830 @tab hyperbolic tangent
5831 @cindex @code{tanh()}
5832 @item @code{asinh(x)}
5833 @tab inverse hyperbolic sine
5834 @cindex @code{asinh()}
5835 @item @code{acosh(x)}
5836 @tab inverse hyperbolic cosine
5837 @cindex @code{acosh()}
5838 @item @code{atanh(x)}
5839 @tab inverse hyperbolic tangent
5840 @cindex @code{atanh()}
5842 @tab exponential function
5843 @cindex @code{exp()}
5845 @tab natural logarithm
5846 @cindex @code{log()}
5847 @item @code{eta(x,y)}
5848 @tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
5849 @cindex @code{eta()}
5852 @cindex @code{Li2()}
5853 @item @code{Li(m, x)}
5854 @tab classical polylogarithm as well as multiple polylogarithm
5856 @item @code{G(a, y)}
5857 @tab multiple polylogarithm
5859 @item @code{G(a, s, y)}
5860 @tab multiple polylogarithm with explicit signs for the imaginary parts
5862 @item @code{S(n, p, x)}
5863 @tab Nielsen's generalized polylogarithm
5865 @item @code{H(m, x)}
5866 @tab harmonic polylogarithm
5868 @item @code{zeta(m)}
5869 @tab Riemann's zeta function as well as multiple zeta value
5870 @cindex @code{zeta()}
5871 @item @code{zeta(m, s)}
5872 @tab alternating Euler sum
5873 @cindex @code{zeta()}
5874 @item @code{zetaderiv(n, x)}
5875 @tab derivatives of Riemann's zeta function
5876 @item @code{tgamma(x)}
5878 @cindex @code{tgamma()}
5879 @cindex gamma function
5880 @item @code{lgamma(x)}
5881 @tab logarithm of gamma function
5882 @cindex @code{lgamma()}
5883 @item @code{beta(x, y)}
5884 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5885 @cindex @code{beta()}
5887 @tab psi (digamma) function
5888 @cindex @code{psi()}
5889 @item @code{psi(n, x)}
5890 @tab derivatives of psi function (polygamma functions)
5891 @item @code{factorial(n)}
5892 @tab factorial function @math{n!}
5893 @cindex @code{factorial()}
5894 @item @code{binomial(n, k)}
5895 @tab binomial coefficients
5896 @cindex @code{binomial()}
5897 @item @code{Order(x)}
5898 @tab order term function in truncated power series
5899 @cindex @code{Order()}
5904 For functions that have a branch cut in the complex plane, GiNaC
5905 follows the conventions of C/C++ for systems that do not support a
5906 signed zero. In particular: the natural logarithm (@code{log}) and
5907 the square root (@code{sqrt}) both have their branch cuts running
5908 along the negative real axis. The @code{asin}, @code{acos}, and
5909 @code{atanh} functions all have two branch cuts starting at +/-1 and
5910 running away towards infinity along the real axis. The @code{atan} and
5911 @code{asinh} functions have two branch cuts starting at +/-i and
5912 running away towards infinity along the imaginary axis. The
5913 @code{acosh} function has one branch cut starting at +1 and running
5914 towards -infinity. These functions are continuous as the branch cut
5915 is approached coming around the finite endpoint of the cut in a
5916 counter clockwise direction.
5919 @subsection Expanding functions
5920 @cindex expand trancedent functions
5921 @cindex @code{expand_options::expand_transcendental}
5922 @cindex @code{expand_options::expand_function_args}
5923 GiNaC knows several expansion laws for trancedent functions, e.g.
5929 @command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
5933 $\log(c*d)=\log(c)+\log(d)$,
5936 @command{log(cd)=log(c)+log(d)}
5945 ). In order to use these rules you need to call @code{expand()} method
5946 with the option @code{expand_options::expand_transcendental}. Another
5947 relevant option is @code{expand_options::expand_function_args}. Their
5948 usage and interaction can be seen from the following example:
5951 symbol x("x"), y("y");
5952 ex e=exp(pow(x+y,2));
5953 cout << e.expand() << endl;
5955 cout << e.expand(expand_options::expand_transcendental) << endl;
5957 cout << e.expand(expand_options::expand_function_args) << endl;
5958 // -> exp(2*x*y+x^2+y^2)
5959 cout << e.expand(expand_options::expand_function_args
5960 | expand_options::expand_transcendental) << endl;
5961 // -> exp(y^2)*exp(2*x*y)*exp(x^2)
5964 If both flags are set (as in the last call), then GiNaC tries to get
5965 the maximal expansion. For example, for the exponent GiNaC firstly expands
5966 the argument and then the function. For the logarithm and absolute value,
5967 GiNaC uses the opposite order: firstly expands the function and then its
5968 argument. Of course, a user can fine-tune this behavior by sequential
5969 calls of several @code{expand()} methods with desired flags.
5971 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5972 @c node-name, next, previous, up
5973 @subsection Multiple polylogarithms
5975 @cindex polylogarithm
5976 @cindex Nielsen's generalized polylogarithm
5977 @cindex harmonic polylogarithm
5978 @cindex multiple zeta value
5979 @cindex alternating Euler sum
5980 @cindex multiple polylogarithm
5982 The multiple polylogarithm is the most generic member of a family of functions,
5983 to which others like the harmonic polylogarithm, Nielsen's generalized
5984 polylogarithm and the multiple zeta value belong.
5985 Each of these functions can also be written as a multiple polylogarithm with specific
5986 parameters. This whole family of functions is therefore often referred to simply as
5987 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5988 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5989 @code{Li} and @code{G} in principle represent the same function, the different
5990 notations are more natural to the series representation or the integral
5991 representation, respectively.
5993 To facilitate the discussion of these functions we distinguish between indices and
5994 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5995 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5997 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5998 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5999 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
6000 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
6001 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
6002 @code{s} is not given, the signs default to +1.
6003 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
6004 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
6005 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
6006 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
6007 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
6009 The functions print in LaTeX format as
6011 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
6017 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
6020 $\zeta(m_1,m_2,\ldots,m_k)$.
6023 @command{\mbox@{Li@}_@{m_1,m_2,...,m_k@}(x_1,x_2,...,x_k)},
6024 @command{\mbox@{S@}_@{n,p@}(x)},
6025 @command{\mbox@{H@}_@{m_1,m_2,...,m_k@}(x)} and
6026 @command{\zeta(m_1,m_2,...,m_k)} (with the dots replaced by actual parameters).
6028 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
6029 are printed with a line above, e.g.
6031 $\zeta(5,\overline{2})$.
6034 @command{\zeta(5,\overline@{2@})}.
6036 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
6038 Definitions and analytical as well as numerical properties of multiple polylogarithms
6039 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
6040 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
6041 except for a few differences which will be explicitly stated in the following.
6043 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
6044 that the indices and arguments are understood to be in the same order as in which they appear in
6045 the series representation. This means
6047 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
6050 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
6053 $\zeta(1,2)$ evaluates to infinity.
6056 @code{Li_@{m_1,m_2,m_3@}(x,1,1) = H_@{m_1,m_2,m_3@}(x)} and
6057 @code{Li_@{2,1@}(1,1) = zeta(2,1) = zeta(3)}, but
6058 @code{zeta(1,2)} evaluates to infinity.
6060 So in comparison to the older ones of the referenced publications the order of
6061 indices and arguments for @code{Li} is reversed.
6063 The functions only evaluate if the indices are integers greater than zero, except for the indices
6064 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
6065 will be interpreted as the sequence of signs for the corresponding indices
6066 @code{m} or the sign of the imaginary part for the
6067 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
6068 @code{zeta(lst@{3,4@}, lst@{-1,1@})} means
6070 $\zeta(\overline{3},4)$
6073 @command{zeta(\overline@{3@},4)}
6076 @code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
6078 $G(a-0\epsilon,b+0\epsilon;c)$.
6081 @command{G(a-0\epsilon,b+0\epsilon;c)}.
6083 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
6084 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
6085 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
6086 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
6087 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
6088 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
6089 evaluates also for negative integers and positive even integers. For example:
6092 > Li(@{3,1@},@{x,1@});
6095 -zeta(@{3,2@},@{-1,-1@})
6100 It is easy to tell for a given function into which other function it can be rewritten, may
6101 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
6102 with negative indices or trailing zeros (the example above gives a hint). Signs can
6103 quickly be messed up, for example. Therefore GiNaC offers a C++ function
6104 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
6105 @code{Li} (@code{eval()} already cares for the possible downgrade):
6108 > convert_H_to_Li(@{0,-2,-1,3@},x);
6109 Li(@{3,1,3@},@{-x,1,-1@})
6110 > convert_H_to_Li(@{2,-1,0@},x);
6111 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
6114 Every function can be numerically evaluated for
6115 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
6116 global variable @code{Digits}:
6121 > evalf(zeta(@{3,1,3,1@}));
6122 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
6125 Note that the convention for arguments on the branch cut in GiNaC as stated above is
6126 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
6128 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6136 In long expressions this helps a lot with debugging, because you can easily spot
6137 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6138 cancellations of divergencies happen.
6140 Useful publications:
6142 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6143 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6145 @cite{Harmonic Polylogarithms},
6146 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6148 @cite{Special Values of Multiple Polylogarithms},
6149 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6151 @cite{Numerical Evaluation of Multiple Polylogarithms},
6152 J.Vollinga, S.Weinzierl, hep-ph/0410259
6154 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6155 @c node-name, next, previous, up
6156 @section Complex expressions
6158 @cindex @code{conjugate()}
6160 For dealing with complex expressions there are the methods
6168 that return respectively the complex conjugate, the real part and the
6169 imaginary part of an expression. Complex conjugation works as expected
6170 for all built-in functions and objects. Taking real and imaginary
6171 parts has not yet been implemented for all built-in functions. In cases where
6172 it is not known how to conjugate or take a real/imaginary part one
6173 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6174 is returned. For instance, in case of a complex symbol @code{x}
6175 (symbols are complex by default), one could not simplify
6176 @code{conjugate(x)}. In the case of strings of gamma matrices,
6177 the @code{conjugate} method takes the Dirac conjugate.
6182 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6186 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6187 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6188 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6189 // -> -gamma5*gamma~b*gamma~a
6193 If you declare your own GiNaC functions and you want to conjugate them, you
6194 will have to supply a specialized conjugation method for them (see
6195 @ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
6196 example). GiNaC does not automatically conjugate user-supplied functions
6197 by conjugating their arguments because this would be incorrect on branch
6198 cuts. Also, specialized methods can be provided to take real and imaginary
6199 parts of user-defined functions.
6201 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6202 @c node-name, next, previous, up
6203 @section Solving linear systems of equations
6204 @cindex @code{lsolve()}
6206 The function @code{lsolve()} provides a convenient wrapper around some
6207 matrix operations that comes in handy when a system of linear equations
6211 ex lsolve(const ex & eqns, const ex & symbols,
6212 unsigned options = solve_algo::automatic);
6215 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6216 @code{relational}) while @code{symbols} is a @code{lst} of
6217 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6220 It returns the @code{lst} of solutions as an expression. As an example,
6221 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6225 symbol a("a"), b("b"), x("x"), y("y");
6226 lst eqns = @{a*x+b*y==3, x-y==b@};
6227 lst vars = @{x, y@};
6228 cout << lsolve(eqns, vars) << endl;
6229 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6232 When the linear equations @code{eqns} are underdetermined, the solution
6233 will contain one or more tautological entries like @code{x==x},
6234 depending on the rank of the system. When they are overdetermined, the
6235 solution will be an empty @code{lst}. Note the third optional parameter
6236 to @code{lsolve()}: it accepts the same parameters as
6237 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6241 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6242 @c node-name, next, previous, up
6243 @section Input and output of expressions
6246 @subsection Expression output
6248 @cindex output of expressions
6250 Expressions can simply be written to any stream:
6255 ex e = 4.5*I+pow(x,2)*3/2;
6256 cout << e << endl; // prints '4.5*I+3/2*x^2'
6260 The default output format is identical to the @command{ginsh} input syntax and
6261 to that used by most computer algebra systems, but not directly pastable
6262 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6263 is printed as @samp{x^2}).
6265 It is possible to print expressions in a number of different formats with
6266 a set of stream manipulators;
6269 std::ostream & dflt(std::ostream & os);
6270 std::ostream & latex(std::ostream & os);
6271 std::ostream & tree(std::ostream & os);
6272 std::ostream & csrc(std::ostream & os);
6273 std::ostream & csrc_float(std::ostream & os);
6274 std::ostream & csrc_double(std::ostream & os);
6275 std::ostream & csrc_cl_N(std::ostream & os);
6276 std::ostream & index_dimensions(std::ostream & os);
6277 std::ostream & no_index_dimensions(std::ostream & os);
6280 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6281 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6282 @code{print_csrc()} functions, respectively.
6285 All manipulators affect the stream state permanently. To reset the output
6286 format to the default, use the @code{dflt} manipulator:
6290 cout << latex; // all output to cout will be in LaTeX format from
6292 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6293 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6294 cout << dflt; // revert to default output format
6295 cout << e << endl; // prints '4.5*I+3/2*x^2'
6299 If you don't want to affect the format of the stream you're working with,
6300 you can output to a temporary @code{ostringstream} like this:
6305 s << latex << e; // format of cout remains unchanged
6306 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6310 @anchor{csrc printing}
6312 @cindex @code{csrc_float}
6313 @cindex @code{csrc_double}
6314 @cindex @code{csrc_cl_N}
6315 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6316 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6317 format that can be directly used in a C or C++ program. The three possible
6318 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6319 classes provided by the CLN library):
6323 cout << "f = " << csrc_float << e << ";\n";
6324 cout << "d = " << csrc_double << e << ";\n";
6325 cout << "n = " << csrc_cl_N << e << ";\n";
6329 The above example will produce (note the @code{x^2} being converted to
6333 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6334 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6335 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6339 The @code{tree} manipulator allows dumping the internal structure of an
6340 expression for debugging purposes:
6351 add, hash=0x0, flags=0x3, nops=2
6352 power, hash=0x0, flags=0x3, nops=2
6353 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6354 2 (numeric), hash=0x6526b0fa, flags=0xf
6355 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6358 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6362 @cindex @code{latex}
6363 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6364 It is rather similar to the default format but provides some braces needed
6365 by LaTeX for delimiting boxes and also converts some common objects to
6366 conventional LaTeX names. It is possible to give symbols a special name for
6367 LaTeX output by supplying it as a second argument to the @code{symbol}
6370 For example, the code snippet
6374 symbol x("x", "\\circ");
6375 ex e = lgamma(x).series(x==0,3);
6376 cout << latex << e << endl;
6383 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6384 +\mathcal@{O@}(\circ^@{3@})
6387 @cindex @code{index_dimensions}
6388 @cindex @code{no_index_dimensions}
6389 Index dimensions are normally hidden in the output. To make them visible, use
6390 the @code{index_dimensions} manipulator. The dimensions will be written in
6391 square brackets behind each index value in the default and LaTeX output
6396 symbol x("x"), y("y");
6397 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6398 ex e = indexed(x, mu) * indexed(y, nu);
6401 // prints 'x~mu*y~nu'
6402 cout << index_dimensions << e << endl;
6403 // prints 'x~mu[4]*y~nu[4]'
6404 cout << no_index_dimensions << e << endl;
6405 // prints 'x~mu*y~nu'
6410 @cindex Tree traversal
6411 If you need any fancy special output format, e.g. for interfacing GiNaC
6412 with other algebra systems or for producing code for different
6413 programming languages, you can always traverse the expression tree yourself:
6416 static void my_print(const ex & e)
6418 if (is_a<function>(e))
6419 cout << ex_to<function>(e).get_name();
6421 cout << ex_to<basic>(e).class_name();
6423 size_t n = e.nops();
6425 for (size_t i=0; i<n; i++) @{
6437 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6445 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6446 symbol(y))),numeric(-2)))
6449 If you need an output format that makes it possible to accurately
6450 reconstruct an expression by feeding the output to a suitable parser or
6451 object factory, you should consider storing the expression in an
6452 @code{archive} object and reading the object properties from there.
6453 See the section on archiving for more information.
6456 @subsection Expression input
6457 @cindex input of expressions
6459 GiNaC provides no way to directly read an expression from a stream because
6460 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6461 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6462 @code{y} you defined in your program and there is no way to specify the
6463 desired symbols to the @code{>>} stream input operator.
6465 Instead, GiNaC lets you read an expression from a stream or a string,
6466 specifying the mapping between the input strings and symbols to be used:
6474 parser reader(table);
6475 ex e = reader("2*x+sin(y)");
6479 The input syntax is the same as that used by @command{ginsh} and the stream
6480 output operator @code{<<}. Matching between the input strings and expressions
6481 is given by @samp{table}. The @samp{table} in this example instructs GiNaC
6482 to substitute any input substring ``x'' with symbol @code{x}. Likewise,
6483 the substring ``y'' will be replaced with symbol @code{y}. It's also possible
6484 to map input (sub)strings to arbitrary expressions:
6490 table["x"] = x+log(y)+1;
6491 parser reader(table);
6492 ex e = reader("5*x^3 - x^2");
6493 // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
6497 If no mapping is specified for a particular string GiNaC will create a symbol
6498 with corresponding name. Later on you can obtain all parser generated symbols
6499 with @code{get_syms()} method:
6504 ex e = reader("2*x+sin(y)");
6505 symtab table = reader.get_syms();
6506 symbol x = ex_to<symbol>(table["x"]);
6507 symbol y = ex_to<symbol>(table["y"]);
6511 Sometimes you might want to prevent GiNaC from inserting these extra symbols
6512 (for example, you want treat an unexpected string in the input as an error).
6517 table["x"] = symbol();
6518 parser reader(table);
6519 parser.strict = true;
6522 e = reader("2*x+sin(y)");
6523 @} catch (parse_error& err) @{
6524 cerr << err.what() << endl;
6525 // prints "unknown symbol "y" in the input"
6530 With this parser, it's also easy to implement interactive GiNaC programs.
6531 When running the following program interactively, remember to send an
6532 EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
6537 #include <stdexcept>
6538 #include <ginac/ginac.h>
6539 using namespace std;
6540 using namespace GiNaC;
6544 cout << "Enter an expression containing 'x': " << flush;
6549 symtab table = reader.get_syms();
6550 symbol x = table.find("x") != table.end() ?
6551 ex_to<symbol>(table["x"]) : symbol("x");
6552 cout << "The derivative of " << e << " with respect to x is ";
6553 cout << e.diff(x) << "." << endl;
6554 @} catch (exception &p) @{
6555 cerr << p.what() << endl;
6560 @subsection Compiling expressions to C function pointers
6561 @cindex compiling expressions
6563 Numerical evaluation of algebraic expressions is seamlessly integrated into
6564 GiNaC by help of the CLN library. While CLN allows for very fast arbitrary
6565 precision numerics, which is more than sufficient for most users, sometimes only
6566 the speed of built-in floating point numbers is fast enough, e.g. for Monte
6567 Carlo integration. The only viable option then is the following: print the
6568 expression in C syntax format, manually add necessary C code, compile that
6569 program and run is as a separate application. This is not only cumbersome and
6570 involves a lot of manual intervention, but it also separates the algebraic and
6571 the numerical evaluation into different execution stages.
6573 GiNaC offers a couple of functions that help to avoid these inconveniences and
6574 problems. The functions automatically perform the printing of a GiNaC expression
6575 and the subsequent compiling of its associated C code. The created object code
6576 is then dynamically linked to the currently running program. A function pointer
6577 to the C function that performs the numerical evaluation is returned and can be
6578 used instantly. This all happens automatically, no user intervention is needed.
6580 The following example demonstrates the use of @code{compile_ex}:
6585 ex myexpr = sin(x) / x;
6588 compile_ex(myexpr, x, fp);
6590 cout << fp(3.2) << endl;
6594 The function @code{compile_ex} is called with the expression to be compiled and
6595 its only free variable @code{x}. Upon successful completion the third parameter
6596 contains a valid function pointer to the corresponding C code module. If called
6597 like in the last line only built-in double precision numerics is involved.
6602 The function pointer has to be defined in advance. GiNaC offers three function
6603 pointer types at the moment:
6606 typedef double (*FUNCP_1P) (double);
6607 typedef double (*FUNCP_2P) (double, double);
6608 typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);
6611 @cindex CUBA library
6612 @cindex Monte Carlo integration
6613 @code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
6614 the correct type to be used with the CUBA library
6615 (@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
6616 parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
6619 For every function pointer type there is a matching @code{compile_ex} available:
6622 void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
6623 const std::string filename = "");
6624 void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
6625 FUNCP_2P& fp, const std::string filename = "");
6626 void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
6627 const std::string filename = "");
6630 When the last parameter @code{filename} is not supplied, @code{compile_ex} will
6631 choose a unique random name for the intermediate source and object files it
6632 produces. On program termination these files will be deleted. If one wishes to
6633 keep the C code and the object files, one can supply the @code{filename}
6634 parameter. The intermediate files will use that filename and will not be
6638 @code{link_ex} is a function that allows to dynamically link an existing object
6639 file and to make it available via a function pointer. This is useful if you
6640 have already used @code{compile_ex} on an expression and want to avoid the
6641 compilation step to be performed over and over again when you restart your
6642 program. The precondition for this is of course, that you have chosen a
6643 filename when you did call @code{compile_ex}. For every above mentioned
6644 function pointer type there exists a corresponding @code{link_ex} function:
6647 void link_ex(const std::string filename, FUNCP_1P& fp);
6648 void link_ex(const std::string filename, FUNCP_2P& fp);
6649 void link_ex(const std::string filename, FUNCP_CUBA& fp);
6652 The complete filename (including the suffix @code{.so}) of the object file has
6659 void unlink_ex(const std::string filename);
6662 is supplied for the rare cases when one wishes to close the dynamically linked
6663 object files directly and have the intermediate files (only if filename has not
6664 been given) deleted. Normally one doesn't need this function, because all the
6665 clean-up will be done automatically upon (regular) program termination.
6667 All the described functions will throw an exception in case they cannot perform
6668 correctly, like for example when writing the file or starting the compiler
6669 fails. Since internally the same printing methods as described in section
6670 @ref{csrc printing} are used, only functions and objects that are available in
6671 standard C will compile successfully (that excludes polylogarithms for example
6672 at the moment). Another precondition for success is, of course, that it must be
6673 possible to evaluate the expression numerically. No free variables despite the
6674 ones supplied to @code{compile_ex} should appear in the expression.
6676 @cindex ginac-excompiler
6677 @code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
6678 compiler and produce the object files. This shell script comes with GiNaC and
6679 will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
6680 (typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
6681 export additional compiler flags via the @env{$CXXFLAGS} variable:
6684 setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
6688 @subsection Archiving
6689 @cindex @code{archive} (class)
6692 GiNaC allows creating @dfn{archives} of expressions which can be stored
6693 to or retrieved from files. To create an archive, you declare an object
6694 of class @code{archive} and archive expressions in it, giving each
6695 expression a unique name:
6699 #include <ginac/ginac.h>
6700 using namespace std;
6701 using namespace GiNaC;
6705 symbol x("x"), y("y"), z("z");
6707 ex foo = sin(x + 2*y) + 3*z + 41;
6711 a.archive_ex(foo, "foo");
6712 a.archive_ex(bar, "the second one");
6716 The archive can then be written to a file:
6720 ofstream out("foobar.gar", ios::binary);
6726 The file @file{foobar.gar} contains all information that is needed to
6727 reconstruct the expressions @code{foo} and @code{bar}. The flag
6728 @code{ios::binary} prevents locales setting of your OS tampers the
6729 archive file structure.
6731 @cindex @command{viewgar}
6732 The tool @command{viewgar} that comes with GiNaC can be used to view
6733 the contents of GiNaC archive files:
6736 $ viewgar foobar.gar
6737 foo = 41+sin(x+2*y)+3*z
6738 the second one = 42+sin(x+2*y)+3*z
6741 The point of writing archive files is of course that they can later be
6747 ifstream in("foobar.gar", ios::binary);
6752 And the stored expressions can be retrieved by their name:
6756 lst syms = @{x, y@};
6758 ex ex1 = a2.unarchive_ex(syms, "foo");
6759 ex ex2 = a2.unarchive_ex(syms, "the second one");
6761 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6762 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6763 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6767 Note that you have to supply a list of the symbols which are to be inserted
6768 in the expressions. Symbols in archives are stored by their name only and
6769 if you don't specify which symbols you have, unarchiving the expression will
6770 create new symbols with that name. E.g. if you hadn't included @code{x} in
6771 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6772 have had no effect because the @code{x} in @code{ex1} would have been a
6773 different symbol than the @code{x} which was defined at the beginning of
6774 the program, although both would appear as @samp{x} when printed.
6776 You can also use the information stored in an @code{archive} object to
6777 output expressions in a format suitable for exact reconstruction. The
6778 @code{archive} and @code{archive_node} classes have a couple of member
6779 functions that let you access the stored properties:
6782 static void my_print2(const archive_node & n)
6785 n.find_string("class", class_name);
6786 cout << class_name << "(";
6788 archive_node::propinfovector p;
6789 n.get_properties(p);
6791 size_t num = p.size();
6792 for (size_t i=0; i<num; i++) @{
6793 const string &name = p[i].name;
6794 if (name == "class")
6796 cout << name << "=";
6798 unsigned count = p[i].count;
6802 for (unsigned j=0; j<count; j++) @{
6803 switch (p[i].type) @{
6804 case archive_node::PTYPE_BOOL: @{
6806 n.find_bool(name, x, j);
6807 cout << (x ? "true" : "false");
6810 case archive_node::PTYPE_UNSIGNED: @{
6812 n.find_unsigned(name, x, j);
6816 case archive_node::PTYPE_STRING: @{
6818 n.find_string(name, x, j);
6819 cout << '\"' << x << '\"';
6822 case archive_node::PTYPE_NODE: @{
6823 const archive_node &x = n.find_ex_node(name, j);
6845 ex e = pow(2, x) - y;
6847 my_print2(ar.get_top_node(0)); cout << endl;
6855 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6856 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6857 overall_coeff=numeric(number="0"))
6860 Be warned, however, that the set of properties and their meaning for each
6861 class may change between GiNaC versions.
6864 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6865 @c node-name, next, previous, up
6866 @chapter Extending GiNaC
6868 By reading so far you should have gotten a fairly good understanding of
6869 GiNaC's design patterns. From here on you should start reading the
6870 sources. All we can do now is issue some recommendations how to tackle
6871 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6872 develop some useful extension please don't hesitate to contact the GiNaC
6873 authors---they will happily incorporate them into future versions.
6876 * What does not belong into GiNaC:: What to avoid.
6877 * Symbolic functions:: Implementing symbolic functions.
6878 * Printing:: Adding new output formats.
6879 * Structures:: Defining new algebraic classes (the easy way).
6880 * Adding classes:: Defining new algebraic classes (the hard way).
6884 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6885 @c node-name, next, previous, up
6886 @section What doesn't belong into GiNaC
6888 @cindex @command{ginsh}
6889 First of all, GiNaC's name must be read literally. It is designed to be
6890 a library for use within C++. The tiny @command{ginsh} accompanying
6891 GiNaC makes this even more clear: it doesn't even attempt to provide a
6892 language. There are no loops or conditional expressions in
6893 @command{ginsh}, it is merely a window into the library for the
6894 programmer to test stuff (or to show off). Still, the design of a
6895 complete CAS with a language of its own, graphical capabilities and all
6896 this on top of GiNaC is possible and is without doubt a nice project for
6899 There are many built-in functions in GiNaC that do not know how to
6900 evaluate themselves numerically to a precision declared at runtime
6901 (using @code{Digits}). Some may be evaluated at certain points, but not
6902 generally. This ought to be fixed. However, doing numerical
6903 computations with GiNaC's quite abstract classes is doomed to be
6904 inefficient. For this purpose, the underlying foundation classes
6905 provided by CLN are much better suited.
6908 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6909 @c node-name, next, previous, up
6910 @section Symbolic functions
6912 The easiest and most instructive way to start extending GiNaC is probably to
6913 create your own symbolic functions. These are implemented with the help of
6914 two preprocessor macros:
6916 @cindex @code{DECLARE_FUNCTION}
6917 @cindex @code{REGISTER_FUNCTION}
6919 DECLARE_FUNCTION_<n>P(<name>)
6920 REGISTER_FUNCTION(<name>, <options>)
6923 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6924 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6925 parameters of type @code{ex} and returns a newly constructed GiNaC
6926 @code{function} object that represents your function.
6928 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6929 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6930 set of options that associate the symbolic function with C++ functions you
6931 provide to implement the various methods such as evaluation, derivative,
6932 series expansion etc. They also describe additional attributes the function
6933 might have, such as symmetry and commutation properties, and a name for
6934 LaTeX output. Multiple options are separated by the member access operator
6935 @samp{.} and can be given in an arbitrary order.
6937 (By the way: in case you are worrying about all the macros above we can
6938 assure you that functions are GiNaC's most macro-intense classes. We have
6939 done our best to avoid macros where we can.)
6941 @subsection A minimal example
6943 Here is an example for the implementation of a function with two arguments
6944 that is not further evaluated:
6947 DECLARE_FUNCTION_2P(myfcn)
6949 REGISTER_FUNCTION(myfcn, dummy())
6952 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6953 in algebraic expressions:
6959 ex e = 2*myfcn(42, 1+3*x) - x;
6961 // prints '2*myfcn(42,1+3*x)-x'
6966 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6967 "no options". A function with no options specified merely acts as a kind of
6968 container for its arguments. It is a pure "dummy" function with no associated
6969 logic (which is, however, sometimes perfectly sufficient).
6971 Let's now have a look at the implementation of GiNaC's cosine function for an
6972 example of how to make an "intelligent" function.
6974 @subsection The cosine function
6976 The GiNaC header file @file{inifcns.h} contains the line
6979 DECLARE_FUNCTION_1P(cos)
6982 which declares to all programs using GiNaC that there is a function @samp{cos}
6983 that takes one @code{ex} as an argument. This is all they need to know to use
6984 this function in expressions.
6986 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6987 is its @code{REGISTER_FUNCTION} line:
6990 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6991 evalf_func(cos_evalf).
6992 derivative_func(cos_deriv).
6993 latex_name("\\cos"));
6996 There are four options defined for the cosine function. One of them
6997 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6998 other three indicate the C++ functions in which the "brains" of the cosine
6999 function are defined.
7001 @cindex @code{hold()}
7003 The @code{eval_func()} option specifies the C++ function that implements
7004 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
7005 the same number of arguments as the associated symbolic function (one in this
7006 case) and returns the (possibly transformed or in some way simplified)
7007 symbolically evaluated function (@xref{Automatic evaluation}, for a description
7008 of the automatic evaluation process). If no (further) evaluation is to take
7009 place, the @code{eval_func()} function must return the original function
7010 with @code{.hold()}, to avoid a potential infinite recursion. If your
7011 symbolic functions produce a segmentation fault or stack overflow when
7012 using them in expressions, you are probably missing a @code{.hold()}
7015 The @code{eval_func()} function for the cosine looks something like this
7016 (actually, it doesn't look like this at all, but it should give you an idea
7020 static ex cos_eval(const ex & x)
7022 if ("x is a multiple of 2*Pi")
7024 else if ("x is a multiple of Pi")
7026 else if ("x is a multiple of Pi/2")
7030 else if ("x has the form 'acos(y)'")
7032 else if ("x has the form 'asin(y)'")
7037 return cos(x).hold();
7041 This function is called every time the cosine is used in a symbolic expression:
7047 // this calls cos_eval(Pi), and inserts its return value into
7048 // the actual expression
7055 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
7056 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
7057 symbolic transformation can be done, the unmodified function is returned
7058 with @code{.hold()}.
7060 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
7061 The user has to call @code{evalf()} for that. This is implemented in a
7065 static ex cos_evalf(const ex & x)
7067 if (is_a<numeric>(x))
7068 return cos(ex_to<numeric>(x));
7070 return cos(x).hold();
7074 Since we are lazy we defer the problem of numeric evaluation to somebody else,
7075 in this case the @code{cos()} function for @code{numeric} objects, which in
7076 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
7077 isn't really needed here, but reminds us that the corresponding @code{eval()}
7078 function would require it in this place.
7080 Differentiation will surely turn up and so we need to tell @code{cos}
7081 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
7082 instance, are then handled automatically by @code{basic::diff} and
7086 static ex cos_deriv(const ex & x, unsigned diff_param)
7092 @cindex product rule
7093 The second parameter is obligatory but uninteresting at this point. It
7094 specifies which parameter to differentiate in a partial derivative in
7095 case the function has more than one parameter, and its main application
7096 is for correct handling of the chain rule.
7098 Derivatives of some functions, for example @code{abs()} and
7099 @code{Order()}, could not be evaluated through the chain rule. In such
7100 cases the full derivative may be specified as shown for @code{Order()}:
7103 static ex Order_expl_derivative(const ex & arg, const symbol & s)
7105 return Order(arg.diff(s));
7109 That is, we need to supply a procedure, which returns the expression of
7110 derivative with respect to the variable @code{s} for the argument
7111 @code{arg}. This procedure need to be registered with the function
7112 through the option @code{expl_derivative_func} (see the next
7113 Subsection). In contrast, a partial derivative, e.g. as was defined for
7114 @code{cos()} above, needs to be registered through the option
7115 @code{derivative_func}.
7117 An implementation of the series expansion is not needed for @code{cos()} as
7118 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
7119 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
7120 the other hand, does have poles and may need to do Laurent expansion:
7123 static ex tan_series(const ex & x, const relational & rel,
7124 int order, unsigned options)
7126 // Find the actual expansion point
7127 const ex x_pt = x.subs(rel);
7129 if ("x_pt is not an odd multiple of Pi/2")
7130 throw do_taylor(); // tell function::series() to do Taylor expansion
7132 // On a pole, expand sin()/cos()
7133 return (sin(x)/cos(x)).series(rel, order+2, options);
7137 The @code{series()} implementation of a function @emph{must} return a
7138 @code{pseries} object, otherwise your code will crash.
7140 @subsection Function options
7142 GiNaC functions understand several more options which are always
7143 specified as @code{.option(params)}. None of them are required, but you
7144 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
7145 is a do-nothing option called @code{dummy()} which you can use to define
7146 functions without any special options.
7149 eval_func(<C++ function>)
7150 evalf_func(<C++ function>)
7151 derivative_func(<C++ function>)
7152 expl_derivative_func(<C++ function>)
7153 series_func(<C++ function>)
7154 conjugate_func(<C++ function>)
7157 These specify the C++ functions that implement symbolic evaluation,
7158 numeric evaluation, partial derivatives, explicit derivative, and series
7159 expansion, respectively. They correspond to the GiNaC methods
7160 @code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
7162 The @code{eval_func()} function needs to use @code{.hold()} if no further
7163 automatic evaluation is desired or possible.
7165 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
7166 expansion, which is correct if there are no poles involved. If the function
7167 has poles in the complex plane, the @code{series_func()} needs to check
7168 whether the expansion point is on a pole and fall back to Taylor expansion
7169 if it isn't. Otherwise, the pole usually needs to be regularized by some
7170 suitable transformation.
7173 latex_name(const string & n)
7176 specifies the LaTeX code that represents the name of the function in LaTeX
7177 output. The default is to put the function name in an @code{\mbox@{@}}.
7180 do_not_evalf_params()
7183 This tells @code{evalf()} to not recursively evaluate the parameters of the
7184 function before calling the @code{evalf_func()}.
7187 set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
7190 This allows you to explicitly specify the commutation properties of the
7191 function (@xref{Non-commutative objects}, for an explanation of
7192 (non)commutativity in GiNaC). For example, with an object of type
7193 @code{return_type_t} created like
7196 return_type_t my_type = make_return_type_t<matrix>();
7199 you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
7200 make GiNaC treat your function like a matrix. By default, functions inherit the
7201 commutation properties of their first argument. The utilized template function
7202 @code{make_return_type_t<>()}
7205 template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
7208 can also be called with an argument specifying the representation label of the
7209 non-commutative function (see section on dirac gamma matrices for more
7213 set_symmetry(const symmetry & s)
7216 specifies the symmetry properties of the function with respect to its
7217 arguments. @xref{Indexed objects}, for an explanation of symmetry
7218 specifications. GiNaC will automatically rearrange the arguments of
7219 symmetric functions into a canonical order.
7221 Sometimes you may want to have finer control over how functions are
7222 displayed in the output. For example, the @code{abs()} function prints
7223 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
7224 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
7228 print_func<C>(<C++ function>)
7231 option which is explained in the next section.
7233 @subsection Functions with a variable number of arguments
7235 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
7236 functions with a fixed number of arguments. Sometimes, though, you may need
7237 to have a function that accepts a variable number of expressions. One way to
7238 accomplish this is to pass variable-length lists as arguments. The
7239 @code{Li()} function uses this method for multiple polylogarithms.
7241 It is also possible to define functions that accept a different number of
7242 parameters under the same function name, such as the @code{psi()} function
7243 which can be called either as @code{psi(z)} (the digamma function) or as
7244 @code{psi(n, z)} (polygamma functions). These are actually two different
7245 functions in GiNaC that, however, have the same name. Defining such
7246 functions is not possible with the macros but requires manually fiddling
7247 with GiNaC internals. If you are interested, please consult the GiNaC source
7248 code for the @code{psi()} function (@file{inifcns.h} and
7249 @file{inifcns_gamma.cpp}).
7252 @node Printing, Structures, Symbolic functions, Extending GiNaC
7253 @c node-name, next, previous, up
7254 @section GiNaC's expression output system
7256 GiNaC allows the output of expressions in a variety of different formats
7257 (@pxref{Input/output}). This section will explain how expression output
7258 is implemented internally, and how to define your own output formats or
7259 change the output format of built-in algebraic objects. You will also want
7260 to read this section if you plan to write your own algebraic classes or
7263 @cindex @code{print_context} (class)
7264 @cindex @code{print_dflt} (class)
7265 @cindex @code{print_latex} (class)
7266 @cindex @code{print_tree} (class)
7267 @cindex @code{print_csrc} (class)
7268 All the different output formats are represented by a hierarchy of classes
7269 rooted in the @code{print_context} class, defined in the @file{print.h}
7274 the default output format
7276 output in LaTeX mathematical mode
7278 a dump of the internal expression structure (for debugging)
7280 the base class for C source output
7281 @item print_csrc_float
7282 C source output using the @code{float} type
7283 @item print_csrc_double
7284 C source output using the @code{double} type
7285 @item print_csrc_cl_N
7286 C source output using CLN types
7289 The @code{print_context} base class provides two public data members:
7301 @code{s} is a reference to the stream to output to, while @code{options}
7302 holds flags and modifiers. Currently, there is only one flag defined:
7303 @code{print_options::print_index_dimensions} instructs the @code{idx} class
7304 to print the index dimension which is normally hidden.
7306 When you write something like @code{std::cout << e}, where @code{e} is
7307 an object of class @code{ex}, GiNaC will construct an appropriate
7308 @code{print_context} object (of a class depending on the selected output
7309 format), fill in the @code{s} and @code{options} members, and call
7311 @cindex @code{print()}
7313 void ex::print(const print_context & c, unsigned level = 0) const;
7316 which in turn forwards the call to the @code{print()} method of the
7317 top-level algebraic object contained in the expression.
7319 Unlike other methods, GiNaC classes don't usually override their
7320 @code{print()} method to implement expression output. Instead, the default
7321 implementation @code{basic::print(c, level)} performs a run-time double
7322 dispatch to a function selected by the dynamic type of the object and the
7323 passed @code{print_context}. To this end, GiNaC maintains a separate method
7324 table for each class, similar to the virtual function table used for ordinary
7325 (single) virtual function dispatch.
7327 The method table contains one slot for each possible @code{print_context}
7328 type, indexed by the (internally assigned) serial number of the type. Slots
7329 may be empty, in which case GiNaC will retry the method lookup with the
7330 @code{print_context} object's parent class, possibly repeating the process
7331 until it reaches the @code{print_context} base class. If there's still no
7332 method defined, the method table of the algebraic object's parent class
7333 is consulted, and so on, until a matching method is found (eventually it
7334 will reach the combination @code{basic/print_context}, which prints the
7335 object's class name enclosed in square brackets).
7337 You can think of the print methods of all the different classes and output
7338 formats as being arranged in a two-dimensional matrix with one axis listing
7339 the algebraic classes and the other axis listing the @code{print_context}
7342 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
7343 to implement printing, but then they won't get any of the benefits of the
7344 double dispatch mechanism (such as the ability for derived classes to
7345 inherit only certain print methods from its parent, or the replacement of
7346 methods at run-time).
7348 @subsection Print methods for classes
7350 The method table for a class is set up either in the definition of the class,
7351 by passing the appropriate @code{print_func<C>()} option to
7352 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7353 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7354 can also be used to override existing methods dynamically.
7356 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7357 be a member function of the class (or one of its parent classes), a static
7358 member function, or an ordinary (global) C++ function. The @code{C} template
7359 parameter specifies the appropriate @code{print_context} type for which the
7360 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7361 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7362 the class is the one being implemented by
7363 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7365 For print methods that are member functions, their first argument must be of
7366 a type convertible to a @code{const C &}, and the second argument must be an
7369 For static members and global functions, the first argument must be of a type
7370 convertible to a @code{const T &}, the second argument must be of a type
7371 convertible to a @code{const C &}, and the third argument must be an
7372 @code{unsigned}. A global function will, of course, not have access to
7373 private and protected members of @code{T}.
7375 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7376 and @code{basic::print()}) is used for proper parenthesizing of the output
7377 (and by @code{print_tree} for proper indentation). It can be used for similar
7378 purposes if you write your own output formats.
7380 The explanations given above may seem complicated, but in practice it's
7381 really simple, as shown in the following example. Suppose that we want to
7382 display exponents in LaTeX output not as superscripts but with little
7383 upwards-pointing arrows. This can be achieved in the following way:
7386 void my_print_power_as_latex(const power & p,
7387 const print_latex & c,
7390 // get the precedence of the 'power' class
7391 unsigned power_prec = p.precedence();
7393 // if the parent operator has the same or a higher precedence
7394 // we need parentheses around the power
7395 if (level >= power_prec)
7398 // print the basis and exponent, each enclosed in braces, and
7399 // separated by an uparrow
7401 p.op(0).print(c, power_prec);
7402 c.s << "@}\\uparrow@{";
7403 p.op(1).print(c, power_prec);
7406 // don't forget the closing parenthesis
7407 if (level >= power_prec)
7413 // a sample expression
7414 symbol x("x"), y("y");
7415 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7417 // switch to LaTeX mode
7420 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7423 // now we replace the method for the LaTeX output of powers with
7425 set_print_func<power, print_latex>(my_print_power_as_latex);
7427 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7438 The first argument of @code{my_print_power_as_latex} could also have been
7439 a @code{const basic &}, the second one a @code{const print_context &}.
7442 The above code depends on @code{mul} objects converting their operands to
7443 @code{power} objects for the purpose of printing.
7446 The output of products including negative powers as fractions is also
7447 controlled by the @code{mul} class.
7450 The @code{power/print_latex} method provided by GiNaC prints square roots
7451 using @code{\sqrt}, but the above code doesn't.
7455 It's not possible to restore a method table entry to its previous or default
7456 value. Once you have called @code{set_print_func()}, you can only override
7457 it with another call to @code{set_print_func()}, but you can't easily go back
7458 to the default behavior again (you can, of course, dig around in the GiNaC
7459 sources, find the method that is installed at startup
7460 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7461 one; that is, after you circumvent the C++ member access control@dots{}).
7463 @subsection Print methods for functions
7465 Symbolic functions employ a print method dispatch mechanism similar to the
7466 one used for classes. The methods are specified with @code{print_func<C>()}
7467 function options. If you don't specify any special print methods, the function
7468 will be printed with its name (or LaTeX name, if supplied), followed by a
7469 comma-separated list of arguments enclosed in parentheses.
7471 For example, this is what GiNaC's @samp{abs()} function is defined like:
7474 static ex abs_eval(const ex & arg) @{ ... @}
7475 static ex abs_evalf(const ex & arg) @{ ... @}
7477 static void abs_print_latex(const ex & arg, const print_context & c)
7479 c.s << "@{|"; arg.print(c); c.s << "|@}";
7482 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7484 c.s << "fabs("; arg.print(c); c.s << ")";
7487 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7488 evalf_func(abs_evalf).
7489 print_func<print_latex>(abs_print_latex).
7490 print_func<print_csrc_float>(abs_print_csrc_float).
7491 print_func<print_csrc_double>(abs_print_csrc_float));
7494 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7495 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7497 There is currently no equivalent of @code{set_print_func()} for functions.
7499 @subsection Adding new output formats
7501 Creating a new output format involves subclassing @code{print_context},
7502 which is somewhat similar to adding a new algebraic class
7503 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7504 that needs to go into the class definition, and a corresponding macro
7505 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7506 Every @code{print_context} class needs to provide a default constructor
7507 and a constructor from an @code{std::ostream} and an @code{unsigned}
7510 Here is an example for a user-defined @code{print_context} class:
7513 class print_myformat : public print_dflt
7515 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7517 print_myformat(std::ostream & os, unsigned opt = 0)
7518 : print_dflt(os, opt) @{@}
7521 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7523 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7526 That's all there is to it. None of the actual expression output logic is
7527 implemented in this class. It merely serves as a selector for choosing
7528 a particular format. The algorithms for printing expressions in the new
7529 format are implemented as print methods, as described above.
7531 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7532 exactly like GiNaC's default output format:
7537 ex e = pow(x, 2) + 1;
7539 // this prints "1+x^2"
7542 // this also prints "1+x^2"
7543 e.print(print_myformat()); cout << endl;
7549 To fill @code{print_myformat} with life, we need to supply appropriate
7550 print methods with @code{set_print_func()}, like this:
7553 // This prints powers with '**' instead of '^'. See the LaTeX output
7554 // example above for explanations.
7555 void print_power_as_myformat(const power & p,
7556 const print_myformat & c,
7559 unsigned power_prec = p.precedence();
7560 if (level >= power_prec)
7562 p.op(0).print(c, power_prec);
7564 p.op(1).print(c, power_prec);
7565 if (level >= power_prec)
7571 // install a new print method for power objects
7572 set_print_func<power, print_myformat>(print_power_as_myformat);
7574 // now this prints "1+x**2"
7575 e.print(print_myformat()); cout << endl;
7577 // but the default format is still "1+x^2"
7583 @node Structures, Adding classes, Printing, Extending GiNaC
7584 @c node-name, next, previous, up
7587 If you are doing some very specialized things with GiNaC, or if you just
7588 need some more organized way to store data in your expressions instead of
7589 anonymous lists, you may want to implement your own algebraic classes.
7590 ('algebraic class' means any class directly or indirectly derived from
7591 @code{basic} that can be used in GiNaC expressions).
7593 GiNaC offers two ways of accomplishing this: either by using the
7594 @code{structure<T>} template class, or by rolling your own class from
7595 scratch. This section will discuss the @code{structure<T>} template which
7596 is easier to use but more limited, while the implementation of custom
7597 GiNaC classes is the topic of the next section. However, you may want to
7598 read both sections because many common concepts and member functions are
7599 shared by both concepts, and it will also allow you to decide which approach
7600 is most suited to your needs.
7602 The @code{structure<T>} template, defined in the GiNaC header file
7603 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7604 or @code{class}) into a GiNaC object that can be used in expressions.
7606 @subsection Example: scalar products
7608 Let's suppose that we need a way to handle some kind of abstract scalar
7609 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7610 product class have to store their left and right operands, which can in turn
7611 be arbitrary expressions. Here is a possible way to represent such a
7612 product in a C++ @code{struct}:
7616 #include <ginac/ginac.h>
7617 using namespace std;
7618 using namespace GiNaC;
7624 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7628 The default constructor is required. Now, to make a GiNaC class out of this
7629 data structure, we need only one line:
7632 typedef structure<sprod_s> sprod;
7635 That's it. This line constructs an algebraic class @code{sprod} which
7636 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7637 expressions like any other GiNaC class:
7641 symbol a("a"), b("b");
7642 ex e = sprod(sprod_s(a, b));
7646 Note the difference between @code{sprod} which is the algebraic class, and
7647 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7648 and @code{right} data members. As shown above, an @code{sprod} can be
7649 constructed from an @code{sprod_s} object.
7651 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7652 you could define a little wrapper function like this:
7655 inline ex make_sprod(ex left, ex right)
7657 return sprod(sprod_s(left, right));
7661 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7662 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7663 @code{get_struct()}:
7667 cout << ex_to<sprod>(e)->left << endl;
7669 cout << ex_to<sprod>(e).get_struct().right << endl;
7674 You only have read access to the members of @code{sprod_s}.
7676 The type definition of @code{sprod} is enough to write your own algorithms
7677 that deal with scalar products, for example:
7682 if (is_a<sprod>(p)) @{
7683 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7684 return make_sprod(sp.right, sp.left);
7695 @subsection Structure output
7697 While the @code{sprod} type is useable it still leaves something to be
7698 desired, most notably proper output:
7703 // -> [structure object]
7707 By default, any structure types you define will be printed as
7708 @samp{[structure object]}. To override this you can either specialize the
7709 template's @code{print()} member function, or specify print methods with
7710 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7711 it's not possible to supply class options like @code{print_func<>()} to
7712 structures, so for a self-contained structure type you need to resort to
7713 overriding the @code{print()} function, which is also what we will do here.
7715 The member functions of GiNaC classes are described in more detail in the
7716 next section, but it shouldn't be hard to figure out what's going on here:
7719 void sprod::print(const print_context & c, unsigned level) const
7721 // tree debug output handled by superclass
7722 if (is_a<print_tree>(c))
7723 inherited::print(c, level);
7725 // get the contained sprod_s object
7726 const sprod_s & sp = get_struct();
7728 // print_context::s is a reference to an ostream
7729 c.s << "<" << sp.left << "|" << sp.right << ">";
7733 Now we can print expressions containing scalar products:
7739 cout << swap_sprod(e) << endl;
7744 @subsection Comparing structures
7746 The @code{sprod} class defined so far still has one important drawback: all
7747 scalar products are treated as being equal because GiNaC doesn't know how to
7748 compare objects of type @code{sprod_s}. This can lead to some confusing
7749 and undesired behavior:
7753 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7755 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7756 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7760 To remedy this, we first need to define the operators @code{==} and @code{<}
7761 for objects of type @code{sprod_s}:
7764 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7766 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7769 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7771 return lhs.left.compare(rhs.left) < 0
7772 ? true : lhs.right.compare(rhs.right) < 0;
7776 The ordering established by the @code{<} operator doesn't have to make any
7777 algebraic sense, but it needs to be well defined. Note that we can't use
7778 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7779 in the implementation of these operators because they would construct
7780 GiNaC @code{relational} objects which in the case of @code{<} do not
7781 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7782 decide which one is algebraically 'less').
7784 Next, we need to change our definition of the @code{sprod} type to let
7785 GiNaC know that an ordering relation exists for the embedded objects:
7788 typedef structure<sprod_s, compare_std_less> sprod;
7791 @code{sprod} objects then behave as expected:
7795 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7796 // -> <a|b>-<a^2|b^2>
7797 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7798 // -> <a|b>+<a^2|b^2>
7799 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7801 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7806 The @code{compare_std_less} policy parameter tells GiNaC to use the
7807 @code{std::less} and @code{std::equal_to} functors to compare objects of
7808 type @code{sprod_s}. By default, these functors forward their work to the
7809 standard @code{<} and @code{==} operators, which we have overloaded.
7810 Alternatively, we could have specialized @code{std::less} and
7811 @code{std::equal_to} for class @code{sprod_s}.
7813 GiNaC provides two other comparison policies for @code{structure<T>}
7814 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7815 which does a bit-wise comparison of the contained @code{T} objects.
7816 This should be used with extreme care because it only works reliably with
7817 built-in integral types, and it also compares any padding (filler bytes of
7818 undefined value) that the @code{T} class might have.
7820 @subsection Subexpressions
7822 Our scalar product class has two subexpressions: the left and right
7823 operands. It might be a good idea to make them accessible via the standard
7824 @code{nops()} and @code{op()} methods:
7827 size_t sprod::nops() const
7832 ex sprod::op(size_t i) const
7836 return get_struct().left;
7838 return get_struct().right;
7840 throw std::range_error("sprod::op(): no such operand");
7845 Implementing @code{nops()} and @code{op()} for container types such as
7846 @code{sprod} has two other nice side effects:
7850 @code{has()} works as expected
7852 GiNaC generates better hash keys for the objects (the default implementation
7853 of @code{calchash()} takes subexpressions into account)
7856 @cindex @code{let_op()}
7857 There is a non-const variant of @code{op()} called @code{let_op()} that
7858 allows replacing subexpressions:
7861 ex & sprod::let_op(size_t i)
7863 // every non-const member function must call this
7864 ensure_if_modifiable();
7868 return get_struct().left;
7870 return get_struct().right;
7872 throw std::range_error("sprod::let_op(): no such operand");
7877 Once we have provided @code{let_op()} we also get @code{subs()} and
7878 @code{map()} for free. In fact, every container class that returns a non-null
7879 @code{nops()} value must either implement @code{let_op()} or provide custom
7880 implementations of @code{subs()} and @code{map()}.
7882 In turn, the availability of @code{map()} enables the recursive behavior of a
7883 couple of other default method implementations, in particular @code{evalf()},
7884 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7885 we probably want to provide our own version of @code{expand()} for scalar
7886 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7887 This is left as an exercise for the reader.
7889 The @code{structure<T>} template defines many more member functions that
7890 you can override by specialization to customize the behavior of your
7891 structures. You are referred to the next section for a description of
7892 some of these (especially @code{eval()}). There is, however, one topic
7893 that shall be addressed here, as it demonstrates one peculiarity of the
7894 @code{structure<T>} template: archiving.
7896 @subsection Archiving structures
7898 If you don't know how the archiving of GiNaC objects is implemented, you
7899 should first read the next section and then come back here. You're back?
7902 To implement archiving for structures it is not enough to provide
7903 specializations for the @code{archive()} member function and the
7904 unarchiving constructor (the @code{unarchive()} function has a default
7905 implementation). You also need to provide a unique name (as a string literal)
7906 for each structure type you define. This is because in GiNaC archives,
7907 the class of an object is stored as a string, the class name.
7909 By default, this class name (as returned by the @code{class_name()} member
7910 function) is @samp{structure} for all structure classes. This works as long
7911 as you have only defined one structure type, but if you use two or more you
7912 need to provide a different name for each by specializing the
7913 @code{get_class_name()} member function. Here is a sample implementation
7914 for enabling archiving of the scalar product type defined above:
7917 const char *sprod::get_class_name() @{ return "sprod"; @}
7919 void sprod::archive(archive_node & n) const
7921 inherited::archive(n);
7922 n.add_ex("left", get_struct().left);
7923 n.add_ex("right", get_struct().right);
7926 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7928 n.find_ex("left", get_struct().left, sym_lst);
7929 n.find_ex("right", get_struct().right, sym_lst);
7933 Note that the unarchiving constructor is @code{sprod::structure} and not
7934 @code{sprod::sprod}, and that we don't need to supply an
7935 @code{sprod::unarchive()} function.
7938 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7939 @c node-name, next, previous, up
7940 @section Adding classes
7942 The @code{structure<T>} template provides an way to extend GiNaC with custom
7943 algebraic classes that is easy to use but has its limitations, the most
7944 severe of which being that you can't add any new member functions to
7945 structures. To be able to do this, you need to write a new class definition
7948 This section will explain how to implement new algebraic classes in GiNaC by
7949 giving the example of a simple 'string' class. After reading this section
7950 you will know how to properly declare a GiNaC class and what the minimum
7951 required member functions are that you have to implement. We only cover the
7952 implementation of a 'leaf' class here (i.e. one that doesn't contain
7953 subexpressions). Creating a container class like, for example, a class
7954 representing tensor products is more involved but this section should give
7955 you enough information so you can consult the source to GiNaC's predefined
7956 classes if you want to implement something more complicated.
7958 @subsection Hierarchy of algebraic classes.
7960 @cindex hierarchy of classes
7961 All algebraic classes (that is, all classes that can appear in expressions)
7962 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7963 @code{basic *} represents a generic pointer to an algebraic class. Working
7964 with such pointers directly is cumbersome (think of memory management), hence
7965 GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
7966 To make such wrapping possible every algebraic class has to implement several
7967 methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
7968 (un)archiving (@pxref{Input/output}) require helper methods too. But don't
7969 worry, most of the work is simplified by the following macros (defined
7970 in @file{registrar.h}):
7972 @item @code{GINAC_DECLARE_REGISTERED_CLASS}
7973 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7974 @item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
7977 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
7978 required for memory management, visitors, printing, and (un)archiving.
7979 It takes the name of the class and its direct superclass as arguments.
7980 The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
7981 the opening brace of the class definition.
7983 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
7984 @code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
7985 members of a class so that printing and (un)archiving works. The
7986 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
7987 the source (at global scope, of course, not inside a function).
7989 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
7990 @code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
7991 options, such as custom printing functions.
7993 @subsection A minimalistic example
7995 Now we will start implementing a new class @code{mystring} that allows
7996 placing character strings in algebraic expressions (this is not very useful,
7997 but it's just an example). This class will be a direct subclass of
7998 @code{basic}. You can use this sample implementation as a starting point
7999 for your own classes @footnote{The self-contained source for this example is
8000 included in GiNaC, see the @file{doc/examples/mystring.cpp} file.}.
8002 The code snippets given here assume that you have included some header files
8008 #include <stdexcept>
8009 #include <ginac/ginac.h>
8010 using namespace std;
8011 using namespace GiNaC;
8014 Now we can write down the class declaration. The class stores a C++
8015 @code{string} and the user shall be able to construct a @code{mystring}
8016 object from a string:
8019 class mystring : public basic
8021 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
8024 mystring(const string & s);
8030 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8033 The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
8034 for memory management, visitors, printing, and (un)archiving.
8035 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
8036 of a class so that printing and (un)archiving works.
8038 Now there are three member functions we have to implement to get a working
8044 @code{mystring()}, the default constructor.
8047 @cindex @code{compare_same_type()}
8048 @code{int compare_same_type(const basic & other)}, which is used internally
8049 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
8050 -1, depending on the relative order of this object and the @code{other}
8051 object. If it returns 0, the objects are considered equal.
8052 @strong{Please notice:} This has nothing to do with the (numeric) ordering
8053 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
8054 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
8055 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
8056 must provide a @code{compare_same_type()} function, even those representing
8057 objects for which no reasonable algebraic ordering relationship can be
8061 And, of course, @code{mystring(const string& s)} which is the constructor
8066 Let's proceed step-by-step. The default constructor looks like this:
8069 mystring::mystring() @{ @}
8072 In the default constructor you should set all other member variables to
8073 reasonable default values (we don't need that here since our @code{str}
8074 member gets set to an empty string automatically).
8076 Our @code{compare_same_type()} function uses a provided function to compare
8080 int mystring::compare_same_type(const basic & other) const
8082 const mystring &o = static_cast<const mystring &>(other);
8083 int cmpval = str.compare(o.str);
8086 else if (cmpval < 0)
8093 Although this function takes a @code{basic &}, it will always be a reference
8094 to an object of exactly the same class (objects of different classes are not
8095 comparable), so the cast is safe. If this function returns 0, the two objects
8096 are considered equal (in the sense that @math{A-B=0}), so you should compare
8097 all relevant member variables.
8099 Now the only thing missing is our constructor:
8102 mystring::mystring(const string& s) : str(s) @{ @}
8105 No surprises here. We set the @code{str} member from the argument.
8107 That's it! We now have a minimal working GiNaC class that can store
8108 strings in algebraic expressions. Let's confirm that the RTTI works:
8111 ex e = mystring("Hello, world!");
8112 cout << is_a<mystring>(e) << endl;
8115 cout << ex_to<basic>(e).class_name() << endl;
8119 Obviously it does. Let's see what the expression @code{e} looks like:
8123 // -> [mystring object]
8126 Hm, not exactly what we expect, but of course the @code{mystring} class
8127 doesn't yet know how to print itself. This can be done either by implementing
8128 the @code{print()} member function, or, preferably, by specifying a
8129 @code{print_func<>()} class option. Let's say that we want to print the string
8130 surrounded by double quotes:
8133 class mystring : public basic
8137 void do_print(const print_context & c, unsigned level = 0) const;
8141 void mystring::do_print(const print_context & c, unsigned level) const
8143 // print_context::s is a reference to an ostream
8144 c.s << '\"' << str << '\"';
8148 The @code{level} argument is only required for container classes to
8149 correctly parenthesize the output.
8151 Now we need to tell GiNaC that @code{mystring} objects should use the
8152 @code{do_print()} member function for printing themselves. For this, we
8156 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
8162 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
8163 print_func<print_context>(&mystring::do_print))
8166 Let's try again to print the expression:
8170 // -> "Hello, world!"
8173 Much better. If we wanted to have @code{mystring} objects displayed in a
8174 different way depending on the output format (default, LaTeX, etc.), we
8175 would have supplied multiple @code{print_func<>()} options with different
8176 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
8177 separated by dots. This is similar to the way options are specified for
8178 symbolic functions. @xref{Printing}, for a more in-depth description of the
8179 way expression output is implemented in GiNaC.
8181 The @code{mystring} class can be used in arbitrary expressions:
8184 e += mystring("GiNaC rulez");
8186 // -> "GiNaC rulez"+"Hello, world!"
8189 (GiNaC's automatic term reordering is in effect here), or even
8192 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
8194 // -> "One string"^(2*sin(-"Another string"+Pi))
8197 Whether this makes sense is debatable but remember that this is only an
8198 example. At least it allows you to implement your own symbolic algorithms
8201 Note that GiNaC's algebraic rules remain unchanged:
8204 e = mystring("Wow") * mystring("Wow");
8208 e = pow(mystring("First")-mystring("Second"), 2);
8209 cout << e.expand() << endl;
8210 // -> -2*"First"*"Second"+"First"^2+"Second"^2
8213 There's no way to, for example, make GiNaC's @code{add} class perform string
8214 concatenation. You would have to implement this yourself.
8216 @subsection Automatic evaluation
8219 @cindex @code{eval()}
8220 @cindex @code{hold()}
8221 When dealing with objects that are just a little more complicated than the
8222 simple string objects we have implemented, chances are that you will want to
8223 have some automatic simplifications or canonicalizations performed on them.
8224 This is done in the evaluation member function @code{eval()}. Let's say that
8225 we wanted all strings automatically converted to lowercase with
8226 non-alphabetic characters stripped, and empty strings removed:
8229 class mystring : public basic
8233 ex eval() const override;
8237 ex mystring::eval() const
8240 for (size_t i=0; i<str.length(); i++) @{
8242 if (c >= 'A' && c <= 'Z')
8243 new_str += tolower(c);
8244 else if (c >= 'a' && c <= 'z')
8248 if (new_str.length() == 0)
8251 return mystring(new_str).hold();
8255 The @code{hold()} member function sets a flag in the object that prevents
8256 further evaluation. Otherwise we might end up in an endless loop. When you
8257 want to return the object unmodified, use @code{return this->hold();}.
8259 If our class had subobjects, we would have to evaluate them first (unless
8260 they are all of type @code{ex}, which are automatically evaluated). We don't
8261 have any subexpressions in the @code{mystring} class, so we are not concerned
8264 Let's confirm that it works:
8267 ex e = mystring("Hello, world!") + mystring("!?#");
8271 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8276 @subsection Optional member functions
8278 We have implemented only a small set of member functions to make the class
8279 work in the GiNaC framework. There are two functions that are not strictly
8280 required but will make operations with objects of the class more efficient:
8282 @cindex @code{calchash()}
8283 @cindex @code{is_equal_same_type()}
8285 unsigned calchash() const override;
8286 bool is_equal_same_type(const basic & other) const override;
8289 The @code{calchash()} method returns an @code{unsigned} hash value for the
8290 object which will allow GiNaC to compare and canonicalize expressions much
8291 more efficiently. You should consult the implementation of some of the built-in
8292 GiNaC classes for examples of hash functions. The default implementation of
8293 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8294 class and all subexpressions that are accessible via @code{op()}.
8296 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8297 tests for equality without establishing an ordering relation, which is often
8298 faster. The default implementation of @code{is_equal_same_type()} just calls
8299 @code{compare_same_type()} and tests its result for zero.
8301 @subsection Other member functions
8303 For a real algebraic class, there are probably some more functions that you
8304 might want to provide:
8307 bool info(unsigned inf) const override;
8308 ex evalf() const override;
8309 ex series(const relational & r, int order, unsigned options = 0) const override;
8310 ex derivative(const symbol & s) const override;
8313 If your class stores sub-expressions (see the scalar product example in the
8314 previous section) you will probably want to override
8316 @cindex @code{let_op()}
8318 size_t nops() const override;
8319 ex op(size_t i) const override;
8320 ex & let_op(size_t i) override;
8321 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
8322 ex map(map_function & f) const override;
8325 @code{let_op()} is a variant of @code{op()} that allows write access. The
8326 default implementations of @code{subs()} and @code{map()} use it, so you have
8327 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8329 You can, of course, also add your own new member functions. Remember
8330 that the RTTI may be used to get information about what kinds of objects
8331 you are dealing with (the position in the class hierarchy) and that you
8332 can always extract the bare object from an @code{ex} by stripping the
8333 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8334 should become a need.
8336 That's it. May the source be with you!
8338 @subsection Upgrading extension classes from older version of GiNaC
8340 GiNaC used to use a custom run time type information system (RTTI). It was
8341 removed from GiNaC. Thus, one needs to rewrite constructors which set
8342 @code{tinfo_key} (which does not exist any more). For example,
8345 myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
8348 needs to be rewritten as
8351 myclass::myclass() @{@}
8354 @node A comparison with other CAS, Advantages, Adding classes, Top
8355 @c node-name, next, previous, up
8356 @chapter A Comparison With Other CAS
8359 This chapter will give you some information on how GiNaC compares to
8360 other, traditional Computer Algebra Systems, like @emph{Maple},
8361 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8362 disadvantages over these systems.
8365 * Advantages:: Strengths of the GiNaC approach.
8366 * Disadvantages:: Weaknesses of the GiNaC approach.
8367 * Why C++?:: Attractiveness of C++.
8370 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8371 @c node-name, next, previous, up
8374 GiNaC has several advantages over traditional Computer
8375 Algebra Systems, like
8380 familiar language: all common CAS implement their own proprietary
8381 grammar which you have to learn first (and maybe learn again when your
8382 vendor decides to `enhance' it). With GiNaC you can write your program
8383 in common C++, which is standardized.
8387 structured data types: you can build up structured data types using
8388 @code{struct}s or @code{class}es together with STL features instead of
8389 using unnamed lists of lists of lists.
8392 strongly typed: in CAS, you usually have only one kind of variables
8393 which can hold contents of an arbitrary type. This 4GL like feature is
8394 nice for novice programmers, but dangerous.
8397 development tools: powerful development tools exist for C++, like fancy
8398 editors (e.g. with automatic indentation and syntax highlighting),
8399 debuggers, visualization tools, documentation generators@dots{}
8402 modularization: C++ programs can easily be split into modules by
8403 separating interface and implementation.
8406 price: GiNaC is distributed under the GNU Public License which means
8407 that it is free and available with source code. And there are excellent
8408 C++-compilers for free, too.
8411 extendable: you can add your own classes to GiNaC, thus extending it on
8412 a very low level. Compare this to a traditional CAS that you can
8413 usually only extend on a high level by writing in the language defined
8414 by the parser. In particular, it turns out to be almost impossible to
8415 fix bugs in a traditional system.
8418 multiple interfaces: Though real GiNaC programs have to be written in
8419 some editor, then be compiled, linked and executed, there are more ways
8420 to work with the GiNaC engine. Many people want to play with
8421 expressions interactively, as in traditional CASs: The tiny
8422 @command{ginsh} that comes with the distribution exposes many, but not
8423 all, of GiNaC's types to a command line.
8426 seamless integration: it is somewhere between difficult and impossible
8427 to call CAS functions from within a program written in C++ or any other
8428 programming language and vice versa. With GiNaC, your symbolic routines
8429 are part of your program. You can easily call third party libraries,
8430 e.g. for numerical evaluation or graphical interaction. All other
8431 approaches are much more cumbersome: they range from simply ignoring the
8432 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8433 system (i.e. @emph{Yacas}).
8436 efficiency: often large parts of a program do not need symbolic
8437 calculations at all. Why use large integers for loop variables or
8438 arbitrary precision arithmetics where @code{int} and @code{double} are
8439 sufficient? For pure symbolic applications, GiNaC is comparable in
8440 speed with other CAS.
8445 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8446 @c node-name, next, previous, up
8447 @section Disadvantages
8449 Of course it also has some disadvantages:
8454 advanced features: GiNaC cannot compete with a program like
8455 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8456 which grows since 1981 by the work of dozens of programmers, with
8457 respect to mathematical features. Integration,
8458 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8459 not planned for the near future).
8462 portability: While the GiNaC library itself is designed to avoid any
8463 platform dependent features (it should compile on any ANSI compliant C++
8464 compiler), the currently used version of the CLN library (fast large
8465 integer and arbitrary precision arithmetics) can only by compiled
8466 without hassle on systems with the C++ compiler from the GNU Compiler
8467 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8468 macros to let the compiler gather all static initializations, which
8469 works for GNU C++ only. Feel free to contact the authors in case you
8470 really believe that you need to use a different compiler. We have
8471 occasionally used other compilers and may be able to give you advice.}
8472 GiNaC uses recent language features like explicit constructors, mutable
8473 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8479 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8480 @c node-name, next, previous, up
8483 Why did we choose to implement GiNaC in C++ instead of Java or any other
8484 language? C++ is not perfect: type checking is not strict (casting is
8485 possible), separation between interface and implementation is not
8486 complete, object oriented design is not enforced. The main reason is
8487 the often scolded feature of operator overloading in C++. While it may
8488 be true that operating on classes with a @code{+} operator is rarely
8489 meaningful, it is perfectly suited for algebraic expressions. Writing
8490 @math{3x+5y} as @code{3*x+5*y} instead of
8491 @code{x.times(3).plus(y.times(5))} looks much more natural.
8492 Furthermore, the main developers are more familiar with C++ than with
8493 any other programming language.
8496 @node Internal structures, Expressions are reference counted, Why C++? , Top
8497 @c node-name, next, previous, up
8498 @appendix Internal structures
8501 * Expressions are reference counted::
8502 * Internal representation of products and sums::
8505 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8506 @c node-name, next, previous, up
8507 @appendixsection Expressions are reference counted
8509 @cindex reference counting
8510 @cindex copy-on-write
8511 @cindex garbage collection
8512 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8513 where the counter belongs to the algebraic objects derived from class
8514 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8515 which @code{ex} contains an instance. If you understood that, you can safely
8516 skip the rest of this passage.
8518 Expressions are extremely light-weight since internally they work like
8519 handles to the actual representation. They really hold nothing more
8520 than a pointer to some other object. What this means in practice is
8521 that whenever you create two @code{ex} and set the second equal to the
8522 first no copying process is involved. Instead, the copying takes place
8523 as soon as you try to change the second. Consider the simple sequence
8528 #include <ginac/ginac.h>
8529 using namespace std;
8530 using namespace GiNaC;
8534 symbol x("x"), y("y"), z("z");
8537 e1 = sin(x + 2*y) + 3*z + 41;
8538 e2 = e1; // e2 points to same object as e1
8539 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8540 e2 += 1; // e2 is copied into a new object
8541 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8545 The line @code{e2 = e1;} creates a second expression pointing to the
8546 object held already by @code{e1}. The time involved for this operation
8547 is therefore constant, no matter how large @code{e1} was. Actual
8548 copying, however, must take place in the line @code{e2 += 1;} because
8549 @code{e1} and @code{e2} are not handles for the same object any more.
8550 This concept is called @dfn{copy-on-write semantics}. It increases
8551 performance considerably whenever one object occurs multiple times and
8552 represents a simple garbage collection scheme because when an @code{ex}
8553 runs out of scope its destructor checks whether other expressions handle
8554 the object it points to too and deletes the object from memory if that
8555 turns out not to be the case. A slightly less trivial example of
8556 differentiation using the chain-rule should make clear how powerful this
8561 symbol x("x"), y("y");
8565 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8566 cout << e1 << endl // prints x+3*y
8567 << e2 << endl // prints (x+3*y)^3
8568 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8572 Here, @code{e1} will actually be referenced three times while @code{e2}
8573 will be referenced two times. When the power of an expression is built,
8574 that expression needs not be copied. Likewise, since the derivative of
8575 a power of an expression can be easily expressed in terms of that
8576 expression, no copying of @code{e1} is involved when @code{e3} is
8577 constructed. So, when @code{e3} is constructed it will print as
8578 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8579 holds a reference to @code{e2} and the factor in front is just
8582 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8583 semantics. When you insert an expression into a second expression, the
8584 result behaves exactly as if the contents of the first expression were
8585 inserted. But it may be useful to remember that this is not what
8586 happens. Knowing this will enable you to write much more efficient
8587 code. If you still have an uncertain feeling with copy-on-write
8588 semantics, we recommend you have a look at the
8589 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8590 Marshall Cline. Chapter 16 covers this issue and presents an
8591 implementation which is pretty close to the one in GiNaC.
8594 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8595 @c node-name, next, previous, up
8596 @appendixsection Internal representation of products and sums
8598 @cindex representation
8601 @cindex @code{power}
8602 Although it should be completely transparent for the user of
8603 GiNaC a short discussion of this topic helps to understand the sources
8604 and also explain performance to a large degree. Consider the
8605 unexpanded symbolic expression
8607 $2d^3 \left( 4a + 5b - 3 \right)$
8610 @math{2*d^3*(4*a+5*b-3)}
8612 which could naively be represented by a tree of linear containers for
8613 addition and multiplication, one container for exponentiation with base
8614 and exponent and some atomic leaves of symbols and numbers in this
8624 @cindex pair-wise representation
8625 However, doing so results in a rather deeply nested tree which will
8626 quickly become inefficient to manipulate. We can improve on this by
8627 representing the sum as a sequence of terms, each one being a pair of a
8628 purely numeric multiplicative coefficient and its rest. In the same
8629 spirit we can store the multiplication as a sequence of terms, each
8630 having a numeric exponent and a possibly complicated base, the tree
8631 becomes much more flat:
8640 The number @code{3} above the symbol @code{d} shows that @code{mul}
8641 objects are treated similarly where the coefficients are interpreted as
8642 @emph{exponents} now. Addition of sums of terms or multiplication of
8643 products with numerical exponents can be coded to be very efficient with
8644 such a pair-wise representation. Internally, this handling is performed
8645 by most CAS in this way. It typically speeds up manipulations by an
8646 order of magnitude. The overall multiplicative factor @code{2} and the
8647 additive term @code{-3} look somewhat out of place in this
8648 representation, however, since they are still carrying a trivial
8649 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8650 this is avoided by adding a field that carries an overall numeric
8651 coefficient. This results in the realistic picture of internal
8654 $2d^3 \left( 4a + 5b - 3 \right)$:
8657 @math{2*d^3*(4*a+5*b-3)}:
8668 This also allows for a better handling of numeric radicals, since
8669 @code{sqrt(2)} can now be carried along calculations. Now it should be
8670 clear, why both classes @code{add} and @code{mul} are derived from the
8671 same abstract class: the data representation is the same, only the
8672 semantics differs. In the class hierarchy, methods for polynomial
8673 expansion and the like are reimplemented for @code{add} and @code{mul},
8674 but the data structure is inherited from @code{expairseq}.
8677 @node Package tools, Configure script options, Internal representation of products and sums, Top
8678 @c node-name, next, previous, up
8679 @appendix Package tools
8681 If you are creating a software package that uses the GiNaC library,
8682 setting the correct command line options for the compiler and linker can
8683 be difficult. The @command{pkg-config} utility makes this process
8684 easier. GiNaC supplies all necessary data in @file{ginac.pc} (installed
8685 into @code{/usr/local/lib/pkgconfig} by default). To compile a simple
8686 program use @footnote{If GiNaC is installed into some non-standard
8687 directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
8688 environment variable to @var{prefix}/lib/pkgconfig for this to work.}
8690 g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
8693 This command line might expand to (for example):
8695 g++ -o simple -lginac -lcln simple.cpp
8698 Not only is the form using @command{pkg-config} easier to type, it will
8699 work on any system, no matter how GiNaC was configured.
8701 For packages configured using GNU automake, @command{pkg-config} also
8702 provides the @code{PKG_CHECK_MODULES} macro to automate the process of
8703 checking for libraries
8706 PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
8707 [@var{ACTION-IF-FOUND}],
8708 [@var{ACTION-IF-NOT-FOUND}])
8716 Determines the location of GiNaC using data from @file{ginac.pc}, which is
8717 either found in the default @command{pkg-config} search path, or from
8718 the environment variable @env{PKG_CONFIG_PATH}.
8721 Tests the installed libraries to make sure that their version
8722 is later than @var{MINIMUM-VERSION}.
8725 If the required version was found, sets the @env{MYAPP_CFLAGS} variable
8726 to the output of @command{pkg-config --cflags ginac} and the @env{MYAPP_LIBS}
8727 variable to the output of @command{pkg-config --libs ginac}, and calls
8728 @samp{AC_SUBST()} for these variables so they can be used in generated
8729 makefiles, and then executes @var{ACTION-IF-FOUND}.
8732 If the required version was not found, executes @var{ACTION-IF-NOT-FOUND}.
8737 * Configure script options:: Configuring a package that uses GiNaC
8738 * Example package:: Example of a package using GiNaC
8742 @node Configure script options, Example package, Package tools, Package tools
8743 @c node-name, next, previous, up
8744 @appendixsection Configuring a package that uses GiNaC
8746 The directory where the GiNaC libraries are installed needs
8747 to be found by your system's dynamic linkers (both compile- and run-time
8748 ones). See the documentation of your system linker for details. Also
8749 make sure that @file{ginac.pc} is in @command{pkg-config}'s search path,
8750 @xref{pkg-config, ,pkg-config, *manpages*}.
8752 The short summary below describes how to do this on a GNU/Linux
8755 Suppose GiNaC is installed into the directory @samp{PREFIX}. To tell
8756 the linkers where to find the library one should
8760 edit @file{/etc/ld.so.conf} and run @command{ldconfig}. For example,
8762 # echo PREFIX/lib >> /etc/ld.so.conf
8767 or set the environment variables @env{LD_LIBRARY_PATH} and @env{LD_RUN_PATH}
8769 $ export LD_LIBRARY_PATH=PREFIX/lib
8770 $ export LD_RUN_PATH=PREFIX/lib
8774 or give a @samp{-L} and @samp{--rpath} flags when running configure,
8778 $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure
8782 To tell @command{pkg-config} where the @file{ginac.pc} file is,
8783 set the @env{PKG_CONFIG_PATH} environment variable:
8785 $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig
8788 Finally, run the @command{configure} script
8793 @c There are many other ways to do the same, @xref{Options, ,Command Line Options, ld, GNU ld manual}.
8795 @node Example package, Bibliography, Configure script options, Package tools
8796 @c node-name, next, previous, up
8797 @appendixsection Example of a package using GiNaC
8799 The following shows how to build a simple package using automake
8800 and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
8804 #include <ginac/ginac.h>
8808 GiNaC::symbol x("x");
8809 GiNaC::ex a = GiNaC::sin(x);
8810 std::cout << "Derivative of " << a
8811 << " is " << a.diff(x) << std::endl;
8816 You should first read the introductory portions of the automake
8817 Manual, if you are not already familiar with it.
8819 Two files are needed, @file{configure.ac}, which is used to build the
8823 dnl Process this file with autoreconf to produce a configure script.
8824 AC_INIT([simple], 1.0.0, bogus@@example.net)
8825 AC_CONFIG_SRCDIR(simple.cpp)
8826 AM_INIT_AUTOMAKE([foreign 1.8])
8832 PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)
8837 The @samp{PKG_CHECK_MODULES} macro does the following: If a GiNaC version
8838 greater or equal than 1.3.7 is found, then it defines @var{SIMPLE_CFLAGS}
8839 and @var{SIMPLE_LIBS}. Otherwise, it dies with the error message like
8841 configure: error: Package requirements (ginac >= 1.3.7) were not met:
8843 Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5
8845 Consider adjusting the PKG_CONFIG_PATH environment variable if you
8846 installed software in a non-standard prefix.
8848 Alternatively, you may set the environment variables SIMPLE_CFLAGS
8849 and SIMPLE_LIBS to avoid the need to call pkg-config.
8850 See the pkg-config man page for more details.
8853 And the @file{Makefile.am}, which will be used to build the Makefile.
8856 ## Process this file with automake to produce Makefile.in
8857 bin_PROGRAMS = simple
8858 simple_SOURCES = simple.cpp
8859 simple_CPPFLAGS = $(SIMPLE_CFLAGS)
8860 simple_LDADD = $(SIMPLE_LIBS)
8863 This @file{Makefile.am}, says that we are building a single executable,
8864 from a single source file @file{simple.cpp}. Since every program
8865 we are building uses GiNaC we could have simply added @var{SIMPLE_CFLAGS}
8866 to @var{CPPFLAGS} and @var{SIMPLE_LIBS} to @var{LIBS}. However, it is
8867 more flexible to specify libraries and complier options on a per-program
8870 To try this example out, create a new directory and add the three
8873 Now execute the following command:
8879 You now have a package that can be built in the normal fashion
8888 @node Bibliography, Concept index, Example package, Top
8889 @c node-name, next, previous, up
8890 @appendix Bibliography
8895 @cite{ISO/IEC 14882:2011: Programming Languages: C++}
8898 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8901 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8904 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8907 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8908 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8911 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8912 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8913 Academic Press, London
8916 @cite{Computer Algebra Systems - A Practical Guide},
8917 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8920 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8921 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8924 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8925 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8928 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8933 @node Concept index, , Bibliography, Top
8934 @c node-name, next, previous, up
8935 @unnumbered Concept index