1 \input texinfo @c -*-texinfo-*-
3 @setfilename ginac.info
4 @settitle GiNaC, an open framework for symbolic computation within the C++ programming language
11 @c I hate putting "@noindent" in front of every paragraph.
19 * ginac: (ginac). C++ library for symbolic computation.
23 This is a tutorial that documents GiNaC @value{VERSION}, an open
24 framework for symbolic computation within the C++ programming language.
26 Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany
28 Permission is granted to make and distribute verbatim copies of
29 this manual provided the copyright notice and this permission notice
30 are preserved on all copies.
33 Permission is granted to process this file through TeX and print the
34 results, provided the printed document carries copying permission
35 notice identical to this one except for the removal of this paragraph
38 Permission is granted to copy and distribute modified versions of this
39 manual under the conditions for verbatim copying, provided that the entire
40 resulting derived work is distributed under the terms of a permission
41 notice identical to this one.
45 @c finalout prevents ugly black rectangles on overfull hbox lines
47 @title GiNaC @value{VERSION}
48 @subtitle An open framework for symbolic computation within the C++ programming language
49 @subtitle @value{UPDATED}
50 @author @uref{http://www.ginac.de}
53 @vskip 0pt plus 1filll
54 Copyright @copyright{} 1999-2006 Johannes Gutenberg University Mainz, Germany
56 Permission is granted to make and distribute verbatim copies of
57 this manual provided the copyright notice and this permission notice
58 are preserved on all copies.
60 Permission is granted to copy and distribute modified versions of this
61 manual under the conditions for verbatim copying, provided that the entire
62 resulting derived work is distributed under the terms of a permission
63 notice identical to this one.
72 @node Top, Introduction, (dir), (dir)
73 @c node-name, next, previous, up
76 This is a tutorial that documents GiNaC @value{VERSION}, an open
77 framework for symbolic computation within the C++ programming language.
80 * Introduction:: GiNaC's purpose.
81 * A tour of GiNaC:: A quick tour of the library.
82 * Installation:: How to install the package.
83 * Basic concepts:: Description of fundamental classes.
84 * Methods and functions:: Algorithms for symbolic manipulations.
85 * Extending GiNaC:: How to extend the library.
86 * A comparison with other CAS:: Compares GiNaC to traditional CAS.
87 * Internal structures:: Description of some internal structures.
88 * Package tools:: Configuring packages to work with GiNaC.
94 @node Introduction, A tour of GiNaC, Top, Top
95 @c node-name, next, previous, up
97 @cindex history of GiNaC
99 The motivation behind GiNaC derives from the observation that most
100 present day computer algebra systems (CAS) are linguistically and
101 semantically impoverished. Although they are quite powerful tools for
102 learning math and solving particular problems they lack modern
103 linguistic structures that allow for the creation of large-scale
104 projects. GiNaC is an attempt to overcome this situation by extending a
105 well established and standardized computer language (C++) by some
106 fundamental symbolic capabilities, thus allowing for integrated systems
107 that embed symbolic manipulations together with more established areas
108 of computer science (like computation-intense numeric applications,
109 graphical interfaces, etc.) under one roof.
111 The particular problem that led to the writing of the GiNaC framework is
112 still a very active field of research, namely the calculation of higher
113 order corrections to elementary particle interactions. There,
114 theoretical physicists are interested in matching present day theories
115 against experiments taking place at particle accelerators. The
116 computations involved are so complex they call for a combined symbolical
117 and numerical approach. This turned out to be quite difficult to
118 accomplish with the present day CAS we have worked with so far and so we
119 tried to fill the gap by writing GiNaC. But of course its applications
120 are in no way restricted to theoretical physics.
122 This tutorial is intended for the novice user who is new to GiNaC but
123 already has some background in C++ programming. However, since a
124 hand-made documentation like this one is difficult to keep in sync with
125 the development, the actual documentation is inside the sources in the
126 form of comments. That documentation may be parsed by one of the many
127 Javadoc-like documentation systems. If you fail at generating it you
128 may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
129 page}. It is an invaluable resource not only for the advanced user who
130 wishes to extend the system (or chase bugs) but for everybody who wants
131 to comprehend the inner workings of GiNaC. This little tutorial on the
132 other hand only covers the basic things that are unlikely to change in
136 The GiNaC framework for symbolic computation within the C++ programming
137 language is Copyright @copyright{} 1999-2006 Johannes Gutenberg
138 University Mainz, Germany.
140 This program is free software; you can redistribute it and/or
141 modify it under the terms of the GNU General Public License as
142 published by the Free Software Foundation; either version 2 of the
143 License, or (at your option) any later version.
145 This program is distributed in the hope that it will be useful, but
146 WITHOUT ANY WARRANTY; without even the implied warranty of
147 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
148 General Public License for more details.
150 You should have received a copy of the GNU General Public License
151 along with this program; see the file COPYING. If not, write to the
152 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
156 @node A tour of GiNaC, How to use it from within C++, Introduction, Top
157 @c node-name, next, previous, up
158 @chapter A Tour of GiNaC
160 This quick tour of GiNaC wants to arise your interest in the
161 subsequent chapters by showing off a bit. Please excuse us if it
162 leaves many open questions.
165 * How to use it from within C++:: Two simple examples.
166 * What it can do for you:: A Tour of GiNaC's features.
170 @node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
171 @c node-name, next, previous, up
172 @section How to use it from within C++
174 The GiNaC open framework for symbolic computation within the C++ programming
175 language does not try to define a language of its own as conventional
176 CAS do. Instead, it extends the capabilities of C++ by symbolic
177 manipulations. Here is how to generate and print a simple (and rather
178 pointless) bivariate polynomial with some large coefficients:
182 #include <ginac/ginac.h>
184 using namespace GiNaC;
188 symbol x("x"), y("y");
191 for (int i=0; i<3; ++i)
192 poly += factorial(i+16)*pow(x,i)*pow(y,2-i);
194 cout << poly << endl;
199 Assuming the file is called @file{hello.cc}, on our system we can compile
200 and run it like this:
203 $ c++ hello.cc -o hello -lcln -lginac
205 355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
208 (@xref{Package tools}, for tools that help you when creating a software
209 package that uses GiNaC.)
211 @cindex Hermite polynomial
212 Next, there is a more meaningful C++ program that calls a function which
213 generates Hermite polynomials in a specified free variable.
217 #include <ginac/ginac.h>
219 using namespace GiNaC;
221 ex HermitePoly(const symbol & x, int n)
223 ex HKer=exp(-pow(x, 2));
224 // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
225 return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
232 for (int i=0; i<6; ++i)
233 cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;
239 When run, this will type out
245 H_3(z) == -12*z+8*z^3
246 H_4(z) == -48*z^2+16*z^4+12
247 H_5(z) == 120*z-160*z^3+32*z^5
250 This method of generating the coefficients is of course far from optimal
251 for production purposes.
253 In order to show some more examples of what GiNaC can do we will now use
254 the @command{ginsh}, a simple GiNaC interactive shell that provides a
255 convenient window into GiNaC's capabilities.
258 @node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
259 @c node-name, next, previous, up
260 @section What it can do for you
262 @cindex @command{ginsh}
263 After invoking @command{ginsh} one can test and experiment with GiNaC's
264 features much like in other Computer Algebra Systems except that it does
265 not provide programming constructs like loops or conditionals. For a
266 concise description of the @command{ginsh} syntax we refer to its
267 accompanied man page. Suffice to say that assignments and comparisons in
268 @command{ginsh} are written as they are in C, i.e. @code{=} assigns and
271 It can manipulate arbitrary precision integers in a very fast way.
272 Rational numbers are automatically converted to fractions of coprime
277 369988485035126972924700782451696644186473100389722973815184405301748249
279 123329495011708990974900260817232214728824366796574324605061468433916083
286 Exact numbers are always retained as exact numbers and only evaluated as
287 floating point numbers if requested. For instance, with numeric
288 radicals is dealt pretty much as with symbols. Products of sums of them
292 > expand((1+a^(1/5)-a^(2/5))^3);
293 1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
294 > expand((1+3^(1/5)-3^(2/5))^3);
296 > evalf((1+3^(1/5)-3^(2/5))^3);
297 0.33408977534118624228
300 The function @code{evalf} that was used above converts any number in
301 GiNaC's expressions into floating point numbers. This can be done to
302 arbitrary predefined accuracy:
306 0.14285714285714285714
310 0.1428571428571428571428571428571428571428571428571428571428571428571428
311 5714285714285714285714285714285714285
314 Exact numbers other than rationals that can be manipulated in GiNaC
315 include predefined constants like Archimedes' @code{Pi}. They can both
316 be used in symbolic manipulations (as an exact number) as well as in
317 numeric expressions (as an inexact number):
323 9.869604401089358619+x
327 11.869604401089358619
330 Built-in functions evaluate immediately to exact numbers if
331 this is possible. Conversions that can be safely performed are done
332 immediately; conversions that are not generally valid are not done:
343 (Note that converting the last input to @code{x} would allow one to
344 conclude that @code{42*Pi} is equal to @code{0}.)
346 Linear equation systems can be solved along with basic linear
347 algebra manipulations over symbolic expressions. In C++ GiNaC offers
348 a matrix class for this purpose but we can see what it can do using
349 @command{ginsh}'s bracket notation to type them in:
352 > lsolve(a+x*y==z,x);
354 > lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
356 > M = [ [1, 3], [-3, 2] ];
360 > charpoly(M,lambda);
362 > A = [ [1, 1], [2, -1] ];
365 [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
368 > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
369 > evalm(B^(2^12345));
370 [[1,0,0],[0,1,0],[0,0,1]]
373 Multivariate polynomials and rational functions may be expanded,
374 collected and normalized (i.e. converted to a ratio of two coprime
378 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
379 12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
380 > b = x^2 + 4*x*y - y^2;
383 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
385 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
387 12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
392 You can differentiate functions and expand them as Taylor or Laurent
393 series in a very natural syntax (the second argument of @code{series} is
394 a relation defining the evaluation point, the third specifies the
397 @cindex Zeta function
401 > series(sin(x),x==0,4);
403 > series(1/tan(x),x==0,4);
404 x^(-1)-1/3*x+Order(x^2)
405 > series(tgamma(x),x==0,3);
406 x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
407 (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
409 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
410 -(0.90747907608088628905)*x^2+Order(x^3)
411 > series(tgamma(2*sin(x)-2),x==Pi/2,6);
412 -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
413 -Euler-1/12+Order((x-1/2*Pi)^3)
416 Here we have made use of the @command{ginsh}-command @code{%} to pop the
417 previously evaluated element from @command{ginsh}'s internal stack.
419 Often, functions don't have roots in closed form. Nevertheless, it's
420 quite easy to compute a solution numerically, to arbitrary precision:
425 > fsolve(cos(x)==x,x,0,2);
426 0.7390851332151606416553120876738734040134117589007574649658
428 > X=fsolve(f,x,-10,10);
429 2.2191071489137460325957851882042901681753665565320678854155
431 -6.372367644529809108115521591070847222364418220770475144296E-58
434 Notice how the final result above differs slightly from zero by about
435 @math{6*10^(-58)}. This is because with 50 decimal digits precision the
436 root cannot be represented more accurately than @code{X}. Such
437 inaccuracies are to be expected when computing with finite floating
440 If you ever wanted to convert units in C or C++ and found this is
441 cumbersome, here is the solution. Symbolic types can always be used as
442 tags for different types of objects. Converting from wrong units to the
443 metric system is now easy:
451 140613.91592783185568*kg*m^(-2)
455 @node Installation, Prerequisites, What it can do for you, Top
456 @c node-name, next, previous, up
457 @chapter Installation
460 GiNaC's installation follows the spirit of most GNU software. It is
461 easily installed on your system by three steps: configuration, build,
465 * Prerequisites:: Packages upon which GiNaC depends.
466 * Configuration:: How to configure GiNaC.
467 * Building GiNaC:: How to compile GiNaC.
468 * Installing GiNaC:: How to install GiNaC on your system.
472 @node Prerequisites, Configuration, Installation, Installation
473 @c node-name, next, previous, up
474 @section Prerequisites
476 In order to install GiNaC on your system, some prerequisites need to be
477 met. First of all, you need to have a C++-compiler adhering to the
478 ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
479 so if you have a different compiler you are on your own. For the
480 configuration to succeed you need a Posix compliant shell installed in
481 @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
482 process as well, since some of the source files are automatically
483 generated by Perl scripts. Last but not least, the CLN library
484 is used extensively and needs to be installed on your system.
485 Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
486 (it is covered by GPL) and install it prior to trying to install
487 GiNaC. The configure script checks if it can find it and if it cannot
488 it will refuse to continue.
491 @node Configuration, Building GiNaC, Prerequisites, Installation
492 @c node-name, next, previous, up
493 @section Configuration
494 @cindex configuration
497 To configure GiNaC means to prepare the source distribution for
498 building. It is done via a shell script called @command{configure} that
499 is shipped with the sources and was originally generated by GNU
500 Autoconf. Since a configure script generated by GNU Autoconf never
501 prompts, all customization must be done either via command line
502 parameters or environment variables. It accepts a list of parameters,
503 the complete set of which can be listed by calling it with the
504 @option{--help} option. The most important ones will be shortly
505 described in what follows:
510 @option{--disable-shared}: When given, this option switches off the
511 build of a shared library, i.e. a @file{.so} file. This may be convenient
512 when developing because it considerably speeds up compilation.
515 @option{--prefix=@var{PREFIX}}: The directory where the compiled library
516 and headers are installed. It defaults to @file{/usr/local} which means
517 that the library is installed in the directory @file{/usr/local/lib},
518 the header files in @file{/usr/local/include/ginac} and the documentation
519 (like this one) into @file{/usr/local/share/doc/GiNaC}.
522 @option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
523 the library installed in some other directory than
524 @file{@var{PREFIX}/lib/}.
527 @option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
528 to have the header files installed in some other directory than
529 @file{@var{PREFIX}/include/ginac/}. For instance, if you specify
530 @option{--includedir=/usr/include} you will end up with the header files
531 sitting in the directory @file{/usr/include/ginac/}. Note that the
532 subdirectory @file{ginac} is enforced by this process in order to
533 keep the header files separated from others. This avoids some
534 clashes and allows for an easier deinstallation of GiNaC. This ought
535 to be considered A Good Thing (tm).
538 @option{--datadir=@var{DATADIR}}: This option may be given in case you
539 want to have the documentation installed in some other directory than
540 @file{@var{PREFIX}/share/doc/GiNaC/}.
544 In addition, you may specify some environment variables. @env{CXX}
545 holds the path and the name of the C++ compiler in case you want to
546 override the default in your path. (The @command{configure} script
547 searches your path for @command{c++}, @command{g++}, @command{gcc},
548 @command{CC}, @command{cxx} and @command{cc++} in that order.) It may
549 be very useful to define some compiler flags with the @env{CXXFLAGS}
550 environment variable, like optimization, debugging information and
551 warning levels. If omitted, it defaults to @option{-g
552 -O2}.@footnote{The @command{configure} script is itself generated from
553 the file @file{configure.ac}. It is only distributed in packaged
554 releases of GiNaC. If you got the naked sources, e.g. from CVS, you
555 must generate @command{configure} along with the various
556 @file{Makefile.in} by using the @command{autogen.sh} script. This will
557 require a fair amount of support from your local toolchain, though.}
559 The whole process is illustrated in the following two
560 examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
561 @command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
564 Here is a simple configuration for a site-wide GiNaC library assuming
565 everything is in default paths:
568 $ export CXXFLAGS="-Wall -O2"
572 And here is a configuration for a private static GiNaC library with
573 several components sitting in custom places (site-wide GCC and private
574 CLN). The compiler is persuaded to be picky and full assertions and
575 debugging information are switched on:
578 $ export CXX=/usr/local/gnu/bin/c++
579 $ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
580 $ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
581 $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
582 $ ./configure --disable-shared --prefix=$(HOME)
586 @node Building GiNaC, Installing GiNaC, Configuration, Installation
587 @c node-name, next, previous, up
588 @section Building GiNaC
589 @cindex building GiNaC
591 After proper configuration you should just build the whole
596 at the command prompt and go for a cup of coffee. The exact time it
597 takes to compile GiNaC depends not only on the speed of your machines
598 but also on other parameters, for instance what value for @env{CXXFLAGS}
599 you entered. Optimization may be very time-consuming.
601 Just to make sure GiNaC works properly you may run a collection of
602 regression tests by typing
608 This will compile some sample programs, run them and check the output
609 for correctness. The regression tests fall in three categories. First,
610 the so called @emph{exams} are performed, simple tests where some
611 predefined input is evaluated (like a pupils' exam). Second, the
612 @emph{checks} test the coherence of results among each other with
613 possible random input. Third, some @emph{timings} are performed, which
614 benchmark some predefined problems with different sizes and display the
615 CPU time used in seconds. Each individual test should return a message
616 @samp{passed}. This is mostly intended to be a QA-check if something
617 was broken during development, not a sanity check of your system. Some
618 of the tests in sections @emph{checks} and @emph{timings} may require
619 insane amounts of memory and CPU time. Feel free to kill them if your
620 machine catches fire. Another quite important intent is to allow people
621 to fiddle around with optimization.
623 By default, the only documentation that will be built is this tutorial
624 in @file{.info} format. To build the GiNaC tutorial and reference manual
625 in HTML, DVI, PostScript, or PDF formats, use one of
634 Generally, the top-level Makefile runs recursively to the
635 subdirectories. It is therefore safe to go into any subdirectory
636 (@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
637 @var{target} there in case something went wrong.
640 @node Installing GiNaC, Basic concepts, Building GiNaC, Installation
641 @c node-name, next, previous, up
642 @section Installing GiNaC
645 To install GiNaC on your system, simply type
651 As described in the section about configuration the files will be
652 installed in the following directories (the directories will be created
653 if they don't already exist):
658 @file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
659 @file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
660 So will @file{libginac.so} unless the configure script was
661 given the option @option{--disable-shared}. The proper symlinks
662 will be established as well.
665 All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
666 (or @file{@var{INCLUDEDIR}/ginac/}, if specified).
669 All documentation (info) will be stuffed into
670 @file{@var{PREFIX}/share/doc/GiNaC/} (or
671 @file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
675 For the sake of completeness we will list some other useful make
676 targets: @command{make clean} deletes all files generated by
677 @command{make}, i.e. all the object files. In addition @command{make
678 distclean} removes all files generated by the configuration and
679 @command{make maintainer-clean} goes one step further and deletes files
680 that may require special tools to rebuild (like the @command{libtool}
681 for instance). Finally @command{make uninstall} removes the installed
682 library, header files and documentation@footnote{Uninstallation does not
683 work after you have called @command{make distclean} since the
684 @file{Makefile} is itself generated by the configuration from
685 @file{Makefile.in} and hence deleted by @command{make distclean}. There
686 are two obvious ways out of this dilemma. First, you can run the
687 configuration again with the same @var{PREFIX} thus creating a
688 @file{Makefile} with a working @samp{uninstall} target. Second, you can
689 do it by hand since you now know where all the files went during
693 @node Basic concepts, Expressions, Installing GiNaC, Top
694 @c node-name, next, previous, up
695 @chapter Basic concepts
697 This chapter will describe the different fundamental objects that can be
698 handled by GiNaC. But before doing so, it is worthwhile introducing you
699 to the more commonly used class of expressions, representing a flexible
700 meta-class for storing all mathematical objects.
703 * Expressions:: The fundamental GiNaC class.
704 * Automatic evaluation:: Evaluation and canonicalization.
705 * Error handling:: How the library reports errors.
706 * The class hierarchy:: Overview of GiNaC's classes.
707 * Symbols:: Symbolic objects.
708 * Numbers:: Numerical objects.
709 * Constants:: Pre-defined constants.
710 * Fundamental containers:: Sums, products and powers.
711 * Lists:: Lists of expressions.
712 * Mathematical functions:: Mathematical functions.
713 * Relations:: Equality, Inequality and all that.
714 * Integrals:: Symbolic integrals.
715 * Matrices:: Matrices.
716 * Indexed objects:: Handling indexed quantities.
717 * Non-commutative objects:: Algebras with non-commutative products.
718 * Hash maps:: A faster alternative to std::map<>.
722 @node Expressions, Automatic evaluation, Basic concepts, Basic concepts
723 @c node-name, next, previous, up
725 @cindex expression (class @code{ex})
728 The most common class of objects a user deals with is the expression
729 @code{ex}, representing a mathematical object like a variable, number,
730 function, sum, product, etc@dots{} Expressions may be put together to form
731 new expressions, passed as arguments to functions, and so on. Here is a
732 little collection of valid expressions:
735 ex MyEx1 = 5; // simple number
736 ex MyEx2 = x + 2*y; // polynomial in x and y
737 ex MyEx3 = (x + 1)/(x - 1); // rational expression
738 ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
739 ex MyEx5 = MyEx4 + 1; // similar to above
742 Expressions are handles to other more fundamental objects, that often
743 contain other expressions thus creating a tree of expressions
744 (@xref{Internal structures}, for particular examples). Most methods on
745 @code{ex} therefore run top-down through such an expression tree. For
746 example, the method @code{has()} scans recursively for occurrences of
747 something inside an expression. Thus, if you have declared @code{MyEx4}
748 as in the example above @code{MyEx4.has(y)} will find @code{y} inside
749 the argument of @code{sin} and hence return @code{true}.
751 The next sections will outline the general picture of GiNaC's class
752 hierarchy and describe the classes of objects that are handled by
755 @subsection Note: Expressions and STL containers
757 GiNaC expressions (@code{ex} objects) have value semantics (they can be
758 assigned, reassigned and copied like integral types) but the operator
759 @code{<} doesn't provide a well-defined ordering on them. In STL-speak,
760 expressions are @samp{Assignable} but not @samp{LessThanComparable}.
762 This implies that in order to use expressions in sorted containers such as
763 @code{std::map<>} and @code{std::set<>} you have to supply a suitable
764 comparison predicate. GiNaC provides such a predicate, called
765 @code{ex_is_less}. For example, a set of expressions should be defined
766 as @code{std::set<ex, ex_is_less>}.
768 Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
769 don't pose a problem. A @code{std::vector<ex>} works as expected.
771 @xref{Information about expressions}, for more about comparing and ordering
775 @node Automatic evaluation, Error handling, Expressions, Basic concepts
776 @c node-name, next, previous, up
777 @section Automatic evaluation and canonicalization of expressions
780 GiNaC performs some automatic transformations on expressions, to simplify
781 them and put them into a canonical form. Some examples:
784 ex MyEx1 = 2*x - 1 + x; // 3*x-1
785 ex MyEx2 = x - x; // 0
786 ex MyEx3 = cos(2*Pi); // 1
787 ex MyEx4 = x*y/x; // y
790 This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
791 evaluation}. GiNaC only performs transformations that are
795 at most of complexity
803 algebraically correct, possibly except for a set of measure zero (e.g.
804 @math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
807 There are two types of automatic transformations in GiNaC that may not
808 behave in an entirely obvious way at first glance:
812 The terms of sums and products (and some other things like the arguments of
813 symmetric functions, the indices of symmetric tensors etc.) are re-ordered
814 into a canonical form that is deterministic, but not lexicographical or in
815 any other way easy to guess (it almost always depends on the number and
816 order of the symbols you define). However, constructing the same expression
817 twice, either implicitly or explicitly, will always result in the same
820 Expressions of the form 'number times sum' are automatically expanded (this
821 has to do with GiNaC's internal representation of sums and products). For
824 ex MyEx5 = 2*(x + y); // 2*x+2*y
825 ex MyEx6 = z*(x + y); // z*(x+y)
829 The general rule is that when you construct expressions, GiNaC automatically
830 creates them in canonical form, which might differ from the form you typed in
831 your program. This may create some awkward looking output (@samp{-y+x} instead
832 of @samp{x-y}) but allows for more efficient operation and usually yields
833 some immediate simplifications.
835 @cindex @code{eval()}
836 Internally, the anonymous evaluator in GiNaC is implemented by the methods
839 ex ex::eval(int level = 0) const;
840 ex basic::eval(int level = 0) const;
843 but unless you are extending GiNaC with your own classes or functions, there
844 should never be any reason to call them explicitly. All GiNaC methods that
845 transform expressions, like @code{subs()} or @code{normal()}, automatically
846 re-evaluate their results.
849 @node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
850 @c node-name, next, previous, up
851 @section Error handling
853 @cindex @code{pole_error} (class)
855 GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
856 generated by GiNaC are subclassed from the standard @code{exception} class
857 defined in the @file{<stdexcept>} header. In addition to the predefined
858 @code{logic_error}, @code{domain_error}, @code{out_of_range},
859 @code{invalid_argument}, @code{runtime_error}, @code{range_error} and
860 @code{overflow_error} types, GiNaC also defines a @code{pole_error}
861 exception that gets thrown when trying to evaluate a mathematical function
864 The @code{pole_error} class has a member function
867 int pole_error::degree() const;
870 that returns the order of the singularity (or 0 when the pole is
871 logarithmic or the order is undefined).
873 When using GiNaC it is useful to arrange for exceptions to be caught in
874 the main program even if you don't want to do any special error handling.
875 Otherwise whenever an error occurs in GiNaC, it will be delegated to the
876 default exception handler of your C++ compiler's run-time system which
877 usually only aborts the program without giving any information what went
880 Here is an example for a @code{main()} function that catches and prints
881 exceptions generated by GiNaC:
886 #include <ginac/ginac.h>
888 using namespace GiNaC;
896 @} catch (exception &p) @{
897 cerr << p.what() << endl;
905 @node The class hierarchy, Symbols, Error handling, Basic concepts
906 @c node-name, next, previous, up
907 @section The class hierarchy
909 GiNaC's class hierarchy consists of several classes representing
910 mathematical objects, all of which (except for @code{ex} and some
911 helpers) are internally derived from one abstract base class called
912 @code{basic}. You do not have to deal with objects of class
913 @code{basic}, instead you'll be dealing with symbols, numbers,
914 containers of expressions and so on.
918 To get an idea about what kinds of symbolic composites may be built we
919 have a look at the most important classes in the class hierarchy and
920 some of the relations among the classes:
922 @image{classhierarchy}
924 The abstract classes shown here (the ones without drop-shadow) are of no
925 interest for the user. They are used internally in order to avoid code
926 duplication if two or more classes derived from them share certain
927 features. An example is @code{expairseq}, a container for a sequence of
928 pairs each consisting of one expression and a number (@code{numeric}).
929 What @emph{is} visible to the user are the derived classes @code{add}
930 and @code{mul}, representing sums and products. @xref{Internal
931 structures}, where these two classes are described in more detail. The
932 following table shortly summarizes what kinds of mathematical objects
933 are stored in the different classes:
936 @multitable @columnfractions .22 .78
937 @item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
938 @item @code{constant} @tab Constants like
945 @item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
946 @item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
947 @item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
948 @item @code{ncmul} @tab Products of non-commutative objects
949 @item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
954 @code{sqrt(}@math{2}@code{)}
957 @item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
958 @item @code{function} @tab A symbolic function like
965 @item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
966 @item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
967 @item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
968 @item @code{indexed} @tab Indexed object like @math{A_ij}
969 @item @code{tensor} @tab Special tensor like the delta and metric tensors
970 @item @code{idx} @tab Index of an indexed object
971 @item @code{varidx} @tab Index with variance
972 @item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
973 @item @code{wildcard} @tab Wildcard for pattern matching
974 @item @code{structure} @tab Template for user-defined classes
979 @node Symbols, Numbers, The class hierarchy, Basic concepts
980 @c node-name, next, previous, up
982 @cindex @code{symbol} (class)
983 @cindex hierarchy of classes
986 Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
987 manipulation what atoms are for chemistry.
989 A typical symbol definition looks like this:
994 This definition actually contains three very different things:
996 @item a C++ variable named @code{x}
997 @item a @code{symbol} object stored in this C++ variable; this object
998 represents the symbol in a GiNaC expression
999 @item the string @code{"x"} which is the name of the symbol, used (almost)
1000 exclusively for printing expressions holding the symbol
1003 Symbols have an explicit name, supplied as a string during construction,
1004 because in C++, variable names can't be used as values, and the C++ compiler
1005 throws them away during compilation.
1007 It is possible to omit the symbol name in the definition:
1012 In this case, GiNaC will assign the symbol an internal, unique name of the
1013 form @code{symbolNNN}. This won't affect the usability of the symbol but
1014 the output of your calculations will become more readable if you give your
1015 symbols sensible names (for intermediate expressions that are only used
1016 internally such anonymous symbols can be quite useful, however).
1018 Now, here is one important property of GiNaC that differentiates it from
1019 other computer algebra programs you may have used: GiNaC does @emph{not} use
1020 the names of symbols to tell them apart, but a (hidden) serial number that
1021 is unique for each newly created @code{symbol} object. In you want to use
1022 one and the same symbol in different places in your program, you must only
1023 create one @code{symbol} object and pass that around. If you create another
1024 symbol, even if it has the same name, GiNaC will treat it as a different
1041 // prints "x^6" which looks right, but...
1043 cout << e.degree(x) << endl;
1044 // ...this doesn't work. The symbol "x" here is different from the one
1045 // in f() and in the expression returned by f(). Consequently, it
1050 One possibility to ensure that @code{f()} and @code{main()} use the same
1051 symbol is to pass the symbol as an argument to @code{f()}:
1053 ex f(int n, const ex & x)
1062 // Now, f() uses the same symbol.
1065 cout << e.degree(x) << endl;
1066 // prints "6", as expected
1070 Another possibility would be to define a global symbol @code{x} that is used
1071 by both @code{f()} and @code{main()}. If you are using global symbols and
1072 multiple compilation units you must take special care, however. Suppose
1073 that you have a header file @file{globals.h} in your program that defines
1074 a @code{symbol x("x");}. In this case, every unit that includes
1075 @file{globals.h} would also get its own definition of @code{x} (because
1076 header files are just inlined into the source code by the C++ preprocessor),
1077 and hence you would again end up with multiple equally-named, but different,
1078 symbols. Instead, the @file{globals.h} header should only contain a
1079 @emph{declaration} like @code{extern symbol x;}, with the definition of
1080 @code{x} moved into a C++ source file such as @file{globals.cpp}.
1082 A different approach to ensuring that symbols used in different parts of
1083 your program are identical is to create them with a @emph{factory} function
1086 const symbol & get_symbol(const string & s)
1088 static map<string, symbol> directory;
1089 map<string, symbol>::iterator i = directory.find(s);
1090 if (i != directory.end())
1093 return directory.insert(make_pair(s, symbol(s))).first->second;
1097 This function returns one newly constructed symbol for each name that is
1098 passed in, and it returns the same symbol when called multiple times with
1099 the same name. Using this symbol factory, we can rewrite our example like
1104 return pow(get_symbol("x"), n);
1111 // Both calls of get_symbol("x") yield the same symbol.
1112 cout << e.degree(get_symbol("x")) << endl;
1117 Instead of creating symbols from strings we could also have
1118 @code{get_symbol()} take, for example, an integer number as its argument.
1119 In this case, we would probably want to give the generated symbols names
1120 that include this number, which can be accomplished with the help of an
1121 @code{ostringstream}.
1123 In general, if you're getting weird results from GiNaC such as an expression
1124 @samp{x-x} that is not simplified to zero, you should check your symbol
1127 As we said, the names of symbols primarily serve for purposes of expression
1128 output. But there are actually two instances where GiNaC uses the names for
1129 identifying symbols: When constructing an expression from a string, and when
1130 recreating an expression from an archive (@pxref{Input/output}).
1132 In addition to its name, a symbol may contain a special string that is used
1135 symbol x("x", "\\Box");
1138 This creates a symbol that is printed as "@code{x}" in normal output, but
1139 as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
1140 information about the different output formats of expressions in GiNaC).
1141 GiNaC automatically creates proper LaTeX code for symbols having names of
1142 greek letters (@samp{alpha}, @samp{mu}, etc.).
1144 @cindex @code{subs()}
1145 Symbols in GiNaC can't be assigned values. If you need to store results of
1146 calculations and give them a name, use C++ variables of type @code{ex}.
1147 If you want to replace a symbol in an expression with something else, you
1148 can invoke the expression's @code{.subs()} method
1149 (@pxref{Substituting expressions}).
1151 @cindex @code{realsymbol()}
1152 By default, symbols are expected to stand in for complex values, i.e. they live
1153 in the complex domain. As a consequence, operations like complex conjugation,
1154 for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
1155 to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
1156 because of the unknown imaginary part of @code{x}.
1157 On the other hand, if you are sure that your symbols will hold only real
1158 values, you would like to have such functions evaluated. Therefore GiNaC
1159 allows you to specify
1160 the domain of the symbol. Instead of @code{symbol x("x");} you can write
1161 @code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.
1163 @cindex @code{possymbol()}
1164 Furthermore, it is also possible to declare a symbol as positive. This will,
1165 for instance, enable the automatic simplification of @code{abs(x)} into
1166 @code{x}. This is done by declaying the symbol as @code{possymbol x("x");}.
1169 @node Numbers, Constants, Symbols, Basic concepts
1170 @c node-name, next, previous, up
1172 @cindex @code{numeric} (class)
1178 For storing numerical things, GiNaC uses Bruno Haible's library CLN.
1179 The classes therein serve as foundation classes for GiNaC. CLN stands
1180 for Class Library for Numbers or alternatively for Common Lisp Numbers.
1181 In order to find out more about CLN's internals, the reader is referred to
1182 the documentation of that library. @inforef{Introduction, , cln}, for
1183 more information. Suffice to say that it is by itself build on top of
1184 another library, the GNU Multiple Precision library GMP, which is an
1185 extremely fast library for arbitrary long integers and rationals as well
1186 as arbitrary precision floating point numbers. It is very commonly used
1187 by several popular cryptographic applications. CLN extends GMP by
1188 several useful things: First, it introduces the complex number field
1189 over either reals (i.e. floating point numbers with arbitrary precision)
1190 or rationals. Second, it automatically converts rationals to integers
1191 if the denominator is unity and complex numbers to real numbers if the
1192 imaginary part vanishes and also correctly treats algebraic functions.
1193 Third it provides good implementations of state-of-the-art algorithms
1194 for all trigonometric and hyperbolic functions as well as for
1195 calculation of some useful constants.
1197 The user can construct an object of class @code{numeric} in several
1198 ways. The following example shows the four most important constructors.
1199 It uses construction from C-integer, construction of fractions from two
1200 integers, construction from C-float and construction from a string:
1204 #include <ginac/ginac.h>
1205 using namespace GiNaC;
1209 numeric two = 2; // exact integer 2
1210 numeric r(2,3); // exact fraction 2/3
1211 numeric e(2.71828); // floating point number
1212 numeric p = "3.14159265358979323846"; // constructor from string
1213 // Trott's constant in scientific notation:
1214 numeric trott("1.0841015122311136151E-2");
1216 std::cout << two*p << std::endl; // floating point 6.283...
1221 @cindex complex numbers
1222 The imaginary unit in GiNaC is a predefined @code{numeric} object with the
1227 numeric z1 = 2-3*I; // exact complex number 2-3i
1228 numeric z2 = 5.9+1.6*I; // complex floating point number
1232 It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
1233 This would, however, call C's built-in operator @code{/} for integers
1234 first and result in a numeric holding a plain integer 1. @strong{Never
1235 use the operator @code{/} on integers} unless you know exactly what you
1236 are doing! Use the constructor from two integers instead, as shown in
1237 the example above. Writing @code{numeric(1)/2} may look funny but works
1240 @cindex @code{Digits}
1242 We have seen now the distinction between exact numbers and floating
1243 point numbers. Clearly, the user should never have to worry about
1244 dynamically created exact numbers, since their `exactness' always
1245 determines how they ought to be handled, i.e. how `long' they are. The
1246 situation is different for floating point numbers. Their accuracy is
1247 controlled by one @emph{global} variable, called @code{Digits}. (For
1248 those readers who know about Maple: it behaves very much like Maple's
1249 @code{Digits}). All objects of class numeric that are constructed from
1250 then on will be stored with a precision matching that number of decimal
1255 #include <ginac/ginac.h>
1256 using namespace std;
1257 using namespace GiNaC;
1261 numeric three(3.0), one(1.0);
1262 numeric x = one/three;
1264 cout << "in " << Digits << " digits:" << endl;
1266 cout << Pi.evalf() << endl;
1278 The above example prints the following output to screen:
1282 0.33333333333333333334
1283 3.1415926535897932385
1285 0.33333333333333333333333333333333333333333333333333333333333333333334
1286 3.1415926535897932384626433832795028841971693993751058209749445923078
1290 Note that the last number is not necessarily rounded as you would
1291 naively expect it to be rounded in the decimal system. But note also,
1292 that in both cases you got a couple of extra digits. This is because
1293 numbers are internally stored by CLN as chunks of binary digits in order
1294 to match your machine's word size and to not waste precision. Thus, on
1295 architectures with different word size, the above output might even
1296 differ with regard to actually computed digits.
1298 It should be clear that objects of class @code{numeric} should be used
1299 for constructing numbers or for doing arithmetic with them. The objects
1300 one deals with most of the time are the polymorphic expressions @code{ex}.
1302 @subsection Tests on numbers
1304 Once you have declared some numbers, assigned them to expressions and
1305 done some arithmetic with them it is frequently desired to retrieve some
1306 kind of information from them like asking whether that number is
1307 integer, rational, real or complex. For those cases GiNaC provides
1308 several useful methods. (Internally, they fall back to invocations of
1309 certain CLN functions.)
1311 As an example, let's construct some rational number, multiply it with
1312 some multiple of its denominator and test what comes out:
1316 #include <ginac/ginac.h>
1317 using namespace std;
1318 using namespace GiNaC;
1320 // some very important constants:
1321 const numeric twentyone(21);
1322 const numeric ten(10);
1323 const numeric five(5);
1327 numeric answer = twentyone;
1330 cout << answer.is_integer() << endl; // false, it's 21/5
1332 cout << answer.is_integer() << endl; // true, it's 42 now!
1336 Note that the variable @code{answer} is constructed here as an integer
1337 by @code{numeric}'s copy constructor but in an intermediate step it
1338 holds a rational number represented as integer numerator and integer
1339 denominator. When multiplied by 10, the denominator becomes unity and
1340 the result is automatically converted to a pure integer again.
1341 Internally, the underlying CLN is responsible for this behavior and we
1342 refer the reader to CLN's documentation. Suffice to say that
1343 the same behavior applies to complex numbers as well as return values of
1344 certain functions. Complex numbers are automatically converted to real
1345 numbers if the imaginary part becomes zero. The full set of tests that
1346 can be applied is listed in the following table.
1349 @multitable @columnfractions .30 .70
1350 @item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
1351 @item @code{.is_zero()}
1352 @tab @dots{}equal to zero
1353 @item @code{.is_positive()}
1354 @tab @dots{}not complex and greater than 0
1355 @item @code{.is_integer()}
1356 @tab @dots{}a (non-complex) integer
1357 @item @code{.is_pos_integer()}
1358 @tab @dots{}an integer and greater than 0
1359 @item @code{.is_nonneg_integer()}
1360 @tab @dots{}an integer and greater equal 0
1361 @item @code{.is_even()}
1362 @tab @dots{}an even integer
1363 @item @code{.is_odd()}
1364 @tab @dots{}an odd integer
1365 @item @code{.is_prime()}
1366 @tab @dots{}a prime integer (probabilistic primality test)
1367 @item @code{.is_rational()}
1368 @tab @dots{}an exact rational number (integers are rational, too)
1369 @item @code{.is_real()}
1370 @tab @dots{}a real integer, rational or float (i.e. is not complex)
1371 @item @code{.is_cinteger()}
1372 @tab @dots{}a (complex) integer (such as @math{2-3*I})
1373 @item @code{.is_crational()}
1374 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
1378 @subsection Numeric functions
1380 The following functions can be applied to @code{numeric} objects and will be
1381 evaluated immediately:
1384 @multitable @columnfractions .30 .70
1385 @item @strong{Name} @tab @strong{Function}
1386 @item @code{inverse(z)}
1387 @tab returns @math{1/z}
1388 @cindex @code{inverse()} (numeric)
1389 @item @code{pow(a, b)}
1390 @tab exponentiation @math{a^b}
1393 @item @code{real(z)}
1395 @cindex @code{real()}
1396 @item @code{imag(z)}
1398 @cindex @code{imag()}
1399 @item @code{csgn(z)}
1400 @tab complex sign (returns an @code{int})
1401 @item @code{step(x)}
1402 @tab step function (returns an @code{numeric})
1403 @item @code{numer(z)}
1404 @tab numerator of rational or complex rational number
1405 @item @code{denom(z)}
1406 @tab denominator of rational or complex rational number
1407 @item @code{sqrt(z)}
1409 @item @code{isqrt(n)}
1410 @tab integer square root
1411 @cindex @code{isqrt()}
1418 @item @code{asin(z)}
1420 @item @code{acos(z)}
1422 @item @code{atan(z)}
1423 @tab inverse tangent
1424 @item @code{atan(y, x)}
1425 @tab inverse tangent with two arguments
1426 @item @code{sinh(z)}
1427 @tab hyperbolic sine
1428 @item @code{cosh(z)}
1429 @tab hyperbolic cosine
1430 @item @code{tanh(z)}
1431 @tab hyperbolic tangent
1432 @item @code{asinh(z)}
1433 @tab inverse hyperbolic sine
1434 @item @code{acosh(z)}
1435 @tab inverse hyperbolic cosine
1436 @item @code{atanh(z)}
1437 @tab inverse hyperbolic tangent
1439 @tab exponential function
1441 @tab natural logarithm
1444 @item @code{zeta(z)}
1445 @tab Riemann's zeta function
1446 @item @code{tgamma(z)}
1448 @item @code{lgamma(z)}
1449 @tab logarithm of gamma function
1451 @tab psi (digamma) function
1452 @item @code{psi(n, z)}
1453 @tab derivatives of psi function (polygamma functions)
1454 @item @code{factorial(n)}
1455 @tab factorial function @math{n!}
1456 @item @code{doublefactorial(n)}
1457 @tab double factorial function @math{n!!}
1458 @cindex @code{doublefactorial()}
1459 @item @code{binomial(n, k)}
1460 @tab binomial coefficients
1461 @item @code{bernoulli(n)}
1462 @tab Bernoulli numbers
1463 @cindex @code{bernoulli()}
1464 @item @code{fibonacci(n)}
1465 @tab Fibonacci numbers
1466 @cindex @code{fibonacci()}
1467 @item @code{mod(a, b)}
1468 @tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
1469 @cindex @code{mod()}
1470 @item @code{smod(a, b)}
1471 @tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
1472 @cindex @code{smod()}
1473 @item @code{irem(a, b)}
1474 @tab integer remainder (has the sign of @math{a}, or is zero)
1475 @cindex @code{irem()}
1476 @item @code{irem(a, b, q)}
1477 @tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
1478 @item @code{iquo(a, b)}
1479 @tab integer quotient
1480 @cindex @code{iquo()}
1481 @item @code{iquo(a, b, r)}
1482 @tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
1483 @item @code{gcd(a, b)}
1484 @tab greatest common divisor
1485 @item @code{lcm(a, b)}
1486 @tab least common multiple
1490 Most of these functions are also available as symbolic functions that can be
1491 used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
1492 as polynomial algorithms.
1494 @subsection Converting numbers
1496 Sometimes it is desirable to convert a @code{numeric} object back to a
1497 built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
1498 class provides a couple of methods for this purpose:
1500 @cindex @code{to_int()}
1501 @cindex @code{to_long()}
1502 @cindex @code{to_double()}
1503 @cindex @code{to_cl_N()}
1505 int numeric::to_int() const;
1506 long numeric::to_long() const;
1507 double numeric::to_double() const;
1508 cln::cl_N numeric::to_cl_N() const;
1511 @code{to_int()} and @code{to_long()} only work when the number they are
1512 applied on is an exact integer. Otherwise the program will halt with a
1513 message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
1514 rational number will return a floating-point approximation. Both
1515 @code{to_int()/to_long()} and @code{to_double()} discard the imaginary
1516 part of complex numbers.
1519 @node Constants, Fundamental containers, Numbers, Basic concepts
1520 @c node-name, next, previous, up
1522 @cindex @code{constant} (class)
1525 @cindex @code{Catalan}
1526 @cindex @code{Euler}
1527 @cindex @code{evalf()}
1528 Constants behave pretty much like symbols except that they return some
1529 specific number when the method @code{.evalf()} is called.
1531 The predefined known constants are:
1534 @multitable @columnfractions .14 .30 .56
1535 @item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
1537 @tab Archimedes' constant
1538 @tab 3.14159265358979323846264338327950288
1539 @item @code{Catalan}
1540 @tab Catalan's constant
1541 @tab 0.91596559417721901505460351493238411
1543 @tab Euler's (or Euler-Mascheroni) constant
1544 @tab 0.57721566490153286060651209008240243
1549 @node Fundamental containers, Lists, Constants, Basic concepts
1550 @c node-name, next, previous, up
1551 @section Sums, products and powers
1555 @cindex @code{power}
1557 Simple rational expressions are written down in GiNaC pretty much like
1558 in other CAS or like expressions involving numerical variables in C.
1559 The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
1560 been overloaded to achieve this goal. When you run the following
1561 code snippet, the constructor for an object of type @code{mul} is
1562 automatically called to hold the product of @code{a} and @code{b} and
1563 then the constructor for an object of type @code{add} is called to hold
1564 the sum of that @code{mul} object and the number one:
1568 symbol a("a"), b("b");
1573 @cindex @code{pow()}
1574 For exponentiation, you have already seen the somewhat clumsy (though C-ish)
1575 statement @code{pow(x,2);} to represent @code{x} squared. This direct
1576 construction is necessary since we cannot safely overload the constructor
1577 @code{^} in C++ to construct a @code{power} object. If we did, it would
1578 have several counterintuitive and undesired effects:
1582 Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
1584 Due to the binding of the operator @code{^}, @code{x^a^b} would result in
1585 @code{(x^a)^b}. This would be confusing since most (though not all) other CAS
1586 interpret this as @code{x^(a^b)}.
1588 Also, expressions involving integer exponents are very frequently used,
1589 which makes it even more dangerous to overload @code{^} since it is then
1590 hard to distinguish between the semantics as exponentiation and the one
1591 for exclusive or. (It would be embarrassing to return @code{1} where one
1592 has requested @code{2^3}.)
1595 @cindex @command{ginsh}
1596 All effects are contrary to mathematical notation and differ from the
1597 way most other CAS handle exponentiation, therefore overloading @code{^}
1598 is ruled out for GiNaC's C++ part. The situation is different in
1599 @command{ginsh}, there the exponentiation-@code{^} exists. (Also note
1600 that the other frequently used exponentiation operator @code{**} does
1601 not exist at all in C++).
1603 To be somewhat more precise, objects of the three classes described
1604 here, are all containers for other expressions. An object of class
1605 @code{power} is best viewed as a container with two slots, one for the
1606 basis, one for the exponent. All valid GiNaC expressions can be
1607 inserted. However, basic transformations like simplifying
1608 @code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
1609 when this is mathematically possible. If we replace the outer exponent
1610 three in the example by some symbols @code{a}, the simplification is not
1611 safe and will not be performed, since @code{a} might be @code{1/2} and
1614 Objects of type @code{add} and @code{mul} are containers with an
1615 arbitrary number of slots for expressions to be inserted. Again, simple
1616 and safe simplifications are carried out like transforming
1617 @code{3*x+4-x} to @code{2*x+4}.
1620 @node Lists, Mathematical functions, Fundamental containers, Basic concepts
1621 @c node-name, next, previous, up
1622 @section Lists of expressions
1623 @cindex @code{lst} (class)
1625 @cindex @code{nops()}
1627 @cindex @code{append()}
1628 @cindex @code{prepend()}
1629 @cindex @code{remove_first()}
1630 @cindex @code{remove_last()}
1631 @cindex @code{remove_all()}
1633 The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
1634 expressions. They are not as ubiquitous as in many other computer algebra
1635 packages, but are sometimes used to supply a variable number of arguments of
1636 the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
1637 constructors, so you should have a basic understanding of them.
1639 Lists can be constructed by assigning a comma-separated sequence of
1644 symbol x("x"), y("y");
1647 // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
1652 There are also constructors that allow direct creation of lists of up to
1653 16 expressions, which is often more convenient but slightly less efficient:
1657 // This produces the same list 'l' as above:
1658 // lst l(x, 2, y, x+y);
1659 // lst l = lst(x, 2, y, x+y);
1663 Use the @code{nops()} method to determine the size (number of expressions) of
1664 a list and the @code{op()} method or the @code{[]} operator to access
1665 individual elements:
1669 cout << l.nops() << endl; // prints '4'
1670 cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
1674 As with the standard @code{list<T>} container, accessing random elements of a
1675 @code{lst} is generally an operation of order @math{O(N)}. Faster read-only
1676 sequential access to the elements of a list is possible with the
1677 iterator types provided by the @code{lst} class:
1680 typedef ... lst::const_iterator;
1681 typedef ... lst::const_reverse_iterator;
1682 lst::const_iterator lst::begin() const;
1683 lst::const_iterator lst::end() const;
1684 lst::const_reverse_iterator lst::rbegin() const;
1685 lst::const_reverse_iterator lst::rend() const;
1688 For example, to print the elements of a list individually you can use:
1693 for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
1698 which is one order faster than
1703 for (size_t i = 0; i < l.nops(); ++i)
1704 cout << l.op(i) << endl;
1708 These iterators also allow you to use some of the algorithms provided by
1709 the C++ standard library:
1713 // print the elements of the list (requires #include <iterator>)
1714 std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
1716 // sum up the elements of the list (requires #include <numeric>)
1717 ex sum = std::accumulate(l.begin(), l.end(), ex(0));
1718 cout << sum << endl; // prints '2+2*x+2*y'
1722 @code{lst} is one of the few GiNaC classes that allow in-place modifications
1723 (the only other one is @code{matrix}). You can modify single elements:
1727 l[1] = 42; // l is now @{x, 42, y, x+y@}
1728 l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
1732 You can append or prepend an expression to a list with the @code{append()}
1733 and @code{prepend()} methods:
1737 l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
1738 l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
1742 You can remove the first or last element of a list with @code{remove_first()}
1743 and @code{remove_last()}:
1747 l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
1748 l.remove_last(); // l is now @{x, 7, y, x+y@}
1752 You can remove all the elements of a list with @code{remove_all()}:
1756 l.remove_all(); // l is now empty
1760 You can bring the elements of a list into a canonical order with @code{sort()}:
1769 // l1 and l2 are now equal
1773 Finally, you can remove all but the first element of consecutive groups of
1774 elements with @code{unique()}:
1779 l3 = x, 2, 2, 2, y, x+y, y+x;
1780 l3.unique(); // l3 is now @{x, 2, y, x+y@}
1785 @node Mathematical functions, Relations, Lists, Basic concepts
1786 @c node-name, next, previous, up
1787 @section Mathematical functions
1788 @cindex @code{function} (class)
1789 @cindex trigonometric function
1790 @cindex hyperbolic function
1792 There are quite a number of useful functions hard-wired into GiNaC. For
1793 instance, all trigonometric and hyperbolic functions are implemented
1794 (@xref{Built-in functions}, for a complete list).
1796 These functions (better called @emph{pseudofunctions}) are all objects
1797 of class @code{function}. They accept one or more expressions as
1798 arguments and return one expression. If the arguments are not
1799 numerical, the evaluation of the function may be halted, as it does in
1800 the next example, showing how a function returns itself twice and
1801 finally an expression that may be really useful:
1803 @cindex Gamma function
1804 @cindex @code{subs()}
1807 symbol x("x"), y("y");
1809 cout << tgamma(foo) << endl;
1810 // -> tgamma(x+(1/2)*y)
1811 ex bar = foo.subs(y==1);
1812 cout << tgamma(bar) << endl;
1814 ex foobar = bar.subs(x==7);
1815 cout << tgamma(foobar) << endl;
1816 // -> (135135/128)*Pi^(1/2)
1820 Besides evaluation most of these functions allow differentiation, series
1821 expansion and so on. Read the next chapter in order to learn more about
1824 It must be noted that these pseudofunctions are created by inline
1825 functions, where the argument list is templated. This means that
1826 whenever you call @code{GiNaC::sin(1)} it is equivalent to
1827 @code{sin(ex(1))} and will therefore not result in a floating point
1828 number. Unless of course the function prototype is explicitly
1829 overridden -- which is the case for arguments of type @code{numeric}
1830 (not wrapped inside an @code{ex}). Hence, in order to obtain a floating
1831 point number of class @code{numeric} you should call
1832 @code{sin(numeric(1))}. This is almost the same as calling
1833 @code{sin(1).evalf()} except that the latter will return a numeric
1834 wrapped inside an @code{ex}.
1837 @node Relations, Integrals, Mathematical functions, Basic concepts
1838 @c node-name, next, previous, up
1840 @cindex @code{relational} (class)
1842 Sometimes, a relation holding between two expressions must be stored
1843 somehow. The class @code{relational} is a convenient container for such
1844 purposes. A relation is by definition a container for two @code{ex} and
1845 a relation between them that signals equality, inequality and so on.
1846 They are created by simply using the C++ operators @code{==}, @code{!=},
1847 @code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
1849 @xref{Mathematical functions}, for examples where various applications
1850 of the @code{.subs()} method show how objects of class relational are
1851 used as arguments. There they provide an intuitive syntax for
1852 substitutions. They are also used as arguments to the @code{ex::series}
1853 method, where the left hand side of the relation specifies the variable
1854 to expand in and the right hand side the expansion point. They can also
1855 be used for creating systems of equations that are to be solved for
1856 unknown variables. But the most common usage of objects of this class
1857 is rather inconspicuous in statements of the form @code{if
1858 (expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
1859 conversion from @code{relational} to @code{bool} takes place. Note,
1860 however, that @code{==} here does not perform any simplifications, hence
1861 @code{expand()} must be called explicitly.
1863 @node Integrals, Matrices, Relations, Basic concepts
1864 @c node-name, next, previous, up
1866 @cindex @code{integral} (class)
1868 An object of class @dfn{integral} can be used to hold a symbolic integral.
1869 If you want to symbolically represent the integral of @code{x*x} from 0 to
1870 1, you would write this as
1872 integral(x, 0, 1, x*x)
1874 The first argument is the integration variable. It should be noted that
1875 GiNaC is not very good (yet?) at symbolically evaluating integrals. In
1876 fact, it can only integrate polynomials. An expression containing integrals
1877 can be evaluated symbolically by calling the
1881 method on it. Numerical evaluation is available by calling the
1885 method on an expression containing the integral. This will only evaluate
1886 integrals into a number if @code{subs}ing the integration variable by a
1887 number in the fourth argument of an integral and then @code{evalf}ing the
1888 result always results in a number. Of course, also the boundaries of the
1889 integration domain must @code{evalf} into numbers. It should be noted that
1890 trying to @code{evalf} a function with discontinuities in the integration
1891 domain is not recommended. The accuracy of the numeric evaluation of
1892 integrals is determined by the static member variable
1894 ex integral::relative_integration_error
1896 of the class @code{integral}. The default value of this is 10^-8.
1897 The integration works by halving the interval of integration, until numeric
1898 stability of the answer indicates that the requested accuracy has been
1899 reached. The maximum depth of the halving can be set via the static member
1902 int integral::max_integration_level
1904 The default value is 15. If this depth is exceeded, @code{evalf} will simply
1905 return the integral unevaluated. The function that performs the numerical
1906 evaluation, is also available as
1908 ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
1911 This function will throw an exception if the maximum depth is exceeded. The
1912 last parameter of the function is optional and defaults to the
1913 @code{relative_integration_error}. To make sure that we do not do too
1914 much work if an expression contains the same integral multiple times,
1915 a lookup table is used.
1917 If you know that an expression holds an integral, you can get the
1918 integration variable, the left boundary, right boundary and integrand by
1919 respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
1920 @code{.op(3)}. Differentiating integrals with respect to variables works
1921 as expected. Note that it makes no sense to differentiate an integral
1922 with respect to the integration variable.
1924 @node Matrices, Indexed objects, Integrals, Basic concepts
1925 @c node-name, next, previous, up
1927 @cindex @code{matrix} (class)
1929 A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
1930 matrix with @math{m} rows and @math{n} columns are accessed with two
1931 @code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
1932 second one in the range 0@dots{}@math{n-1}.
1934 There are a couple of ways to construct matrices, with or without preset
1935 elements. The constructor
1938 matrix::matrix(unsigned r, unsigned c);
1941 creates a matrix with @samp{r} rows and @samp{c} columns with all elements
1944 The fastest way to create a matrix with preinitialized elements is to assign
1945 a list of comma-separated expressions to an empty matrix (see below for an
1946 example). But you can also specify the elements as a (flat) list with
1949 matrix::matrix(unsigned r, unsigned c, const lst & l);
1954 @cindex @code{lst_to_matrix()}
1956 ex lst_to_matrix(const lst & l);
1959 constructs a matrix from a list of lists, each list representing a matrix row.
1961 There is also a set of functions for creating some special types of
1964 @cindex @code{diag_matrix()}
1965 @cindex @code{unit_matrix()}
1966 @cindex @code{symbolic_matrix()}
1968 ex diag_matrix(const lst & l);
1969 ex unit_matrix(unsigned x);
1970 ex unit_matrix(unsigned r, unsigned c);
1971 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
1972 ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
1973 const string & tex_base_name);
1976 @code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
1977 elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
1978 by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
1979 matrix filled with newly generated symbols made of the specified base name
1980 and the position of each element in the matrix.
1982 Matrices often arise by omitting elements of another matrix. For
1983 instance, the submatrix @code{S} of a matrix @code{M} takes a
1984 rectangular block from @code{M}. The reduced matrix @code{R} is defined
1985 by removing one row and one column from a matrix @code{M}. (The
1986 determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
1987 can be used for computing the inverse using Cramer's rule.)
1989 @cindex @code{sub_matrix()}
1990 @cindex @code{reduced_matrix()}
1992 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
1993 ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
1996 The function @code{sub_matrix()} takes a row offset @code{r} and a
1997 column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
1998 columns. The function @code{reduced_matrix()} has two integer arguments
1999 that specify which row and column to remove:
2007 cout << reduced_matrix(m, 1, 1) << endl;
2008 // -> [[11,13],[31,33]]
2009 cout << sub_matrix(m, 1, 2, 1, 2) << endl;
2010 // -> [[22,23],[32,33]]
2014 Matrix elements can be accessed and set using the parenthesis (function call)
2018 const ex & matrix::operator()(unsigned r, unsigned c) const;
2019 ex & matrix::operator()(unsigned r, unsigned c);
2022 It is also possible to access the matrix elements in a linear fashion with
2023 the @code{op()} method. But C++-style subscripting with square brackets
2024 @samp{[]} is not available.
2026 Here are a couple of examples for constructing matrices:
2030 symbol a("a"), b("b");
2044 cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
2047 cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
2050 cout << diag_matrix(lst(a, b)) << endl;
2053 cout << unit_matrix(3) << endl;
2054 // -> [[1,0,0],[0,1,0],[0,0,1]]
2056 cout << symbolic_matrix(2, 3, "x") << endl;
2057 // -> [[x00,x01,x02],[x10,x11,x12]]
2061 @cindex @code{is_zero_matrix()}
2062 The method @code{matrix::is_zero_matrix()} returns @code{true} only if
2063 all entries of the matrix are zeros. There is also method
2064 @code{ex::is_zero_matrix()} which returns @code{true} only if the
2065 expression is zero or a zero matrix.
2067 @cindex @code{transpose()}
2068 There are three ways to do arithmetic with matrices. The first (and most
2069 direct one) is to use the methods provided by the @code{matrix} class:
2072 matrix matrix::add(const matrix & other) const;
2073 matrix matrix::sub(const matrix & other) const;
2074 matrix matrix::mul(const matrix & other) const;
2075 matrix matrix::mul_scalar(const ex & other) const;
2076 matrix matrix::pow(const ex & expn) const;
2077 matrix matrix::transpose() const;
2080 All of these methods return the result as a new matrix object. Here is an
2081 example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
2086 matrix A(2, 2), B(2, 2), C(2, 2);
2094 matrix result = A.mul(B).sub(C.mul_scalar(2));
2095 cout << result << endl;
2096 // -> [[-13,-6],[1,2]]
2101 @cindex @code{evalm()}
2102 The second (and probably the most natural) way is to construct an expression
2103 containing matrices with the usual arithmetic operators and @code{pow()}.
2104 For efficiency reasons, expressions with sums, products and powers of
2105 matrices are not automatically evaluated in GiNaC. You have to call the
2109 ex ex::evalm() const;
2112 to obtain the result:
2119 // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
2120 cout << e.evalm() << endl;
2121 // -> [[-13,-6],[1,2]]
2126 The non-commutativity of the product @code{A*B} in this example is
2127 automatically recognized by GiNaC. There is no need to use a special
2128 operator here. @xref{Non-commutative objects}, for more information about
2129 dealing with non-commutative expressions.
2131 Finally, you can work with indexed matrices and call @code{simplify_indexed()}
2132 to perform the arithmetic:
2137 idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
2138 e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
2140 // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
2141 cout << e.simplify_indexed() << endl;
2142 // -> [[-13,-6],[1,2]].i.j
2146 Using indices is most useful when working with rectangular matrices and
2147 one-dimensional vectors because you don't have to worry about having to
2148 transpose matrices before multiplying them. @xref{Indexed objects}, for
2149 more information about using matrices with indices, and about indices in
2152 The @code{matrix} class provides a couple of additional methods for
2153 computing determinants, traces, characteristic polynomials and ranks:
2155 @cindex @code{determinant()}
2156 @cindex @code{trace()}
2157 @cindex @code{charpoly()}
2158 @cindex @code{rank()}
2160 ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
2161 ex matrix::trace() const;
2162 ex matrix::charpoly(const ex & lambda) const;
2163 unsigned matrix::rank() const;
2166 The @samp{algo} argument of @code{determinant()} allows to select
2167 between different algorithms for calculating the determinant. The
2168 asymptotic speed (as parametrized by the matrix size) can greatly differ
2169 between those algorithms, depending on the nature of the matrix'
2170 entries. The possible values are defined in the @file{flags.h} header
2171 file. By default, GiNaC uses a heuristic to automatically select an
2172 algorithm that is likely (but not guaranteed) to give the result most
2175 @cindex @code{inverse()} (matrix)
2176 @cindex @code{solve()}
2177 Matrices may also be inverted using the @code{ex matrix::inverse()}
2178 method and linear systems may be solved with:
2181 matrix matrix::solve(const matrix & vars, const matrix & rhs,
2182 unsigned algo=solve_algo::automatic) const;
2185 Assuming the matrix object this method is applied on is an @code{m}
2186 times @code{n} matrix, then @code{vars} must be a @code{n} times
2187 @code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
2188 times @code{p} matrix. The returned matrix then has dimension @code{n}
2189 times @code{p} and in the case of an underdetermined system will still
2190 contain some of the indeterminates from @code{vars}. If the system is
2191 overdetermined, an exception is thrown.
2194 @node Indexed objects, Non-commutative objects, Matrices, Basic concepts
2195 @c node-name, next, previous, up
2196 @section Indexed objects
2198 GiNaC allows you to handle expressions containing general indexed objects in
2199 arbitrary spaces. It is also able to canonicalize and simplify such
2200 expressions and perform symbolic dummy index summations. There are a number
2201 of predefined indexed objects provided, like delta and metric tensors.
2203 There are few restrictions placed on indexed objects and their indices and
2204 it is easy to construct nonsense expressions, but our intention is to
2205 provide a general framework that allows you to implement algorithms with
2206 indexed quantities, getting in the way as little as possible.
2208 @cindex @code{idx} (class)
2209 @cindex @code{indexed} (class)
2210 @subsection Indexed quantities and their indices
2212 Indexed expressions in GiNaC are constructed of two special types of objects,
2213 @dfn{index objects} and @dfn{indexed objects}.
2217 @cindex contravariant
2220 @item Index objects are of class @code{idx} or a subclass. Every index has
2221 a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
2222 the index lives in) which can both be arbitrary expressions but are usually
2223 a number or a simple symbol. In addition, indices of class @code{varidx} have
2224 a @dfn{variance} (they can be co- or contravariant), and indices of class
2225 @code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
2227 @item Indexed objects are of class @code{indexed} or a subclass. They
2228 contain a @dfn{base expression} (which is the expression being indexed), and
2229 one or more indices.
2233 @strong{Please notice:} when printing expressions, covariant indices and indices
2234 without variance are denoted @samp{.i} while contravariant indices are
2235 denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
2236 value. In the following, we are going to use that notation in the text so
2237 instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
2238 not visible in the output.
2240 A simple example shall illustrate the concepts:
2244 #include <ginac/ginac.h>
2245 using namespace std;
2246 using namespace GiNaC;
2250 symbol i_sym("i"), j_sym("j");
2251 idx i(i_sym, 3), j(j_sym, 3);
2254 cout << indexed(A, i, j) << endl;
2256 cout << index_dimensions << indexed(A, i, j) << endl;
2258 cout << dflt; // reset cout to default output format (dimensions hidden)
2262 The @code{idx} constructor takes two arguments, the index value and the
2263 index dimension. First we define two index objects, @code{i} and @code{j},
2264 both with the numeric dimension 3. The value of the index @code{i} is the
2265 symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
2266 @code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
2267 construct an expression containing one indexed object, @samp{A.i.j}. It has
2268 the symbol @code{A} as its base expression and the two indices @code{i} and
2271 The dimensions of indices are normally not visible in the output, but one
2272 can request them to be printed with the @code{index_dimensions} manipulator,
2275 Note the difference between the indices @code{i} and @code{j} which are of
2276 class @code{idx}, and the index values which are the symbols @code{i_sym}
2277 and @code{j_sym}. The indices of indexed objects cannot directly be symbols
2278 or numbers but must be index objects. For example, the following is not
2279 correct and will raise an exception:
2282 symbol i("i"), j("j");
2283 e = indexed(A, i, j); // ERROR: indices must be of type idx
2286 You can have multiple indexed objects in an expression, index values can
2287 be numeric, and index dimensions symbolic:
2291 symbol B("B"), dim("dim");
2292 cout << 4 * indexed(A, i)
2293 + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
2298 @code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
2299 index of value 2, and a symbolic index @samp{i} with the symbolic dimension
2300 @samp{dim}. Note that GiNaC doesn't automatically notify you that the free
2301 indices of @samp{A} and @samp{B} in the sum don't match (you have to call
2302 @code{simplify_indexed()} for that, see below).
2304 In fact, base expressions, index values and index dimensions can be
2305 arbitrary expressions:
2309 cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
2314 It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
2315 get an error message from this but you will probably not be able to do
2316 anything useful with it.
2318 @cindex @code{get_value()}
2319 @cindex @code{get_dimension()}
2323 ex idx::get_value();
2324 ex idx::get_dimension();
2327 return the value and dimension of an @code{idx} object. If you have an index
2328 in an expression, such as returned by calling @code{.op()} on an indexed
2329 object, you can get a reference to the @code{idx} object with the function
2330 @code{ex_to<idx>()} on the expression.
2332 There are also the methods
2335 bool idx::is_numeric();
2336 bool idx::is_symbolic();
2337 bool idx::is_dim_numeric();
2338 bool idx::is_dim_symbolic();
2341 for checking whether the value and dimension are numeric or symbolic
2342 (non-numeric). Using the @code{info()} method of an index (see @ref{Information
2343 about expressions}) returns information about the index value.
2345 @cindex @code{varidx} (class)
2346 If you need co- and contravariant indices, use the @code{varidx} class:
2350 symbol mu_sym("mu"), nu_sym("nu");
2351 varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
2352 varidx mu_co(mu_sym, 4, true); // covariant index .mu
2354 cout << indexed(A, mu, nu) << endl;
2356 cout << indexed(A, mu_co, nu) << endl;
2358 cout << indexed(A, mu.toggle_variance(), nu) << endl;
2363 A @code{varidx} is an @code{idx} with an additional flag that marks it as
2364 co- or contravariant. The default is a contravariant (upper) index, but
2365 this can be overridden by supplying a third argument to the @code{varidx}
2366 constructor. The two methods
2369 bool varidx::is_covariant();
2370 bool varidx::is_contravariant();
2373 allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
2374 to get the object reference from an expression). There's also the very useful
2378 ex varidx::toggle_variance();
2381 which makes a new index with the same value and dimension but the opposite
2382 variance. By using it you only have to define the index once.
2384 @cindex @code{spinidx} (class)
2385 The @code{spinidx} class provides dotted and undotted variant indices, as
2386 used in the Weyl-van-der-Waerden spinor formalism:
2390 symbol K("K"), C_sym("C"), D_sym("D");
2391 spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
2392 // contravariant, undotted
2393 spinidx C_co(C_sym, 2, true); // covariant index
2394 spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
2395 spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
2397 cout << indexed(K, C, D) << endl;
2399 cout << indexed(K, C_co, D_dot) << endl;
2401 cout << indexed(K, D_co_dot, D) << endl;
2406 A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
2407 dotted or undotted. The default is undotted but this can be overridden by
2408 supplying a fourth argument to the @code{spinidx} constructor. The two
2412 bool spinidx::is_dotted();
2413 bool spinidx::is_undotted();
2416 allow you to check whether or not a @code{spinidx} object is dotted (use
2417 @code{ex_to<spinidx>()} to get the object reference from an expression).
2418 Finally, the two methods
2421 ex spinidx::toggle_dot();
2422 ex spinidx::toggle_variance_dot();
2425 create a new index with the same value and dimension but opposite dottedness
2426 and the same or opposite variance.
2428 @subsection Substituting indices
2430 @cindex @code{subs()}
2431 Sometimes you will want to substitute one symbolic index with another
2432 symbolic or numeric index, for example when calculating one specific element
2433 of a tensor expression. This is done with the @code{.subs()} method, as it
2434 is done for symbols (see @ref{Substituting expressions}).
2436 You have two possibilities here. You can either substitute the whole index
2437 by another index or expression:
2441 ex e = indexed(A, mu_co);
2442 cout << e << " becomes " << e.subs(mu_co == nu) << endl;
2443 // -> A.mu becomes A~nu
2444 cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
2445 // -> A.mu becomes A~0
2446 cout << e << " becomes " << e.subs(mu_co == 0) << endl;
2447 // -> A.mu becomes A.0
2451 The third example shows that trying to replace an index with something that
2452 is not an index will substitute the index value instead.
2454 Alternatively, you can substitute the @emph{symbol} of a symbolic index by
2459 ex e = indexed(A, mu_co);
2460 cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
2461 // -> A.mu becomes A.nu
2462 cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
2463 // -> A.mu becomes A.0
2467 As you see, with the second method only the value of the index will get
2468 substituted. Its other properties, including its dimension, remain unchanged.
2469 If you want to change the dimension of an index you have to substitute the
2470 whole index by another one with the new dimension.
2472 Finally, substituting the base expression of an indexed object works as
2477 ex e = indexed(A, mu_co);
2478 cout << e << " becomes " << e.subs(A == A+B) << endl;
2479 // -> A.mu becomes (B+A).mu
2483 @subsection Symmetries
2484 @cindex @code{symmetry} (class)
2485 @cindex @code{sy_none()}
2486 @cindex @code{sy_symm()}
2487 @cindex @code{sy_anti()}
2488 @cindex @code{sy_cycl()}
2490 Indexed objects can have certain symmetry properties with respect to their
2491 indices. Symmetries are specified as a tree of objects of class @code{symmetry}
2492 that is constructed with the helper functions
2495 symmetry sy_none(...);
2496 symmetry sy_symm(...);
2497 symmetry sy_anti(...);
2498 symmetry sy_cycl(...);
2501 @code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
2502 specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
2503 represents a cyclic symmetry. Each of these functions accepts up to four
2504 arguments which can be either symmetry objects themselves or unsigned integer
2505 numbers that represent an index position (counting from 0). A symmetry
2506 specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
2507 or @code{sy_cycl()} with no arguments specifies the respective symmetry for
2510 Here are some examples of symmetry definitions:
2515 e = indexed(A, i, j);
2516 e = indexed(A, sy_none(), i, j); // equivalent
2517 e = indexed(A, sy_none(0, 1), i, j); // equivalent
2519 // Symmetric in all three indices:
2520 e = indexed(A, sy_symm(), i, j, k);
2521 e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
2522 e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
2523 // different canonical order
2525 // Symmetric in the first two indices only:
2526 e = indexed(A, sy_symm(0, 1), i, j, k);
2527 e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
2529 // Antisymmetric in the first and last index only (index ranges need not
2531 e = indexed(A, sy_anti(0, 2), i, j, k);
2532 e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
2534 // An example of a mixed symmetry: antisymmetric in the first two and
2535 // last two indices, symmetric when swapping the first and last index
2536 // pairs (like the Riemann curvature tensor):
2537 e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
2539 // Cyclic symmetry in all three indices:
2540 e = indexed(A, sy_cycl(), i, j, k);
2541 e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
2543 // The following examples are invalid constructions that will throw
2544 // an exception at run time.
2546 // An index may not appear multiple times:
2547 e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
2548 e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
2550 // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
2551 // same number of indices:
2552 e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
2554 // And of course, you cannot specify indices which are not there:
2555 e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
2559 If you need to specify more than four indices, you have to use the
2560 @code{.add()} method of the @code{symmetry} class. For example, to specify
2561 full symmetry in the first six indices you would write
2562 @code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
2564 If an indexed object has a symmetry, GiNaC will automatically bring the
2565 indices into a canonical order which allows for some immediate simplifications:
2569 cout << indexed(A, sy_symm(), i, j)
2570 + indexed(A, sy_symm(), j, i) << endl;
2572 cout << indexed(B, sy_anti(), i, j)
2573 + indexed(B, sy_anti(), j, i) << endl;
2575 cout << indexed(B, sy_anti(), i, j, k)
2576 - indexed(B, sy_anti(), j, k, i) << endl;
2581 @cindex @code{get_free_indices()}
2583 @subsection Dummy indices
2585 GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
2586 that a summation over the index range is implied. Symbolic indices which are
2587 not dummy indices are called @dfn{free indices}. Numeric indices are neither
2588 dummy nor free indices.
2590 To be recognized as a dummy index pair, the two indices must be of the same
2591 class and their value must be the same single symbol (an index like
2592 @samp{2*n+1} is never a dummy index). If the indices are of class
2593 @code{varidx} they must also be of opposite variance; if they are of class
2594 @code{spinidx} they must be both dotted or both undotted.
2596 The method @code{.get_free_indices()} returns a vector containing the free
2597 indices of an expression. It also checks that the free indices of the terms
2598 of a sum are consistent:
2602 symbol A("A"), B("B"), C("C");
2604 symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
2605 idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
2607 ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
2608 cout << exprseq(e.get_free_indices()) << endl;
2610 // 'j' and 'l' are dummy indices
2612 symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
2613 varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
2615 e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
2616 + indexed(C, mu, sigma, rho, sigma.toggle_variance());
2617 cout << exprseq(e.get_free_indices()) << endl;
2619 // 'nu' is a dummy index, but 'sigma' is not
2621 e = indexed(A, mu, mu);
2622 cout << exprseq(e.get_free_indices()) << endl;
2624 // 'mu' is not a dummy index because it appears twice with the same
2627 e = indexed(A, mu, nu) + 42;
2628 cout << exprseq(e.get_free_indices()) << endl; // ERROR
2629 // this will throw an exception:
2630 // "add::get_free_indices: inconsistent indices in sum"
2634 @cindex @code{expand_dummy_sum()}
2635 A dummy index summation like
2642 can be expanded for indices with numeric
2643 dimensions (e.g. 3) into the explicit sum like
2645 $a_1b^1+a_2b^2+a_3b^3 $.
2648 a.1 b~1 + a.2 b~2 + a.3 b~3.
2650 This is performed by the function
2653 ex expand_dummy_sum(const ex & e, bool subs_idx = false);
2656 which takes an expression @code{e} and returns the expanded sum for all
2657 dummy indices with numeric dimensions. If the parameter @code{subs_idx}
2658 is set to @code{true} then all substitutions are made by @code{idx} class
2659 indices, i.e. without variance. In this case the above sum
2668 $a_1b_1+a_2b_2+a_3b_3 $.
2671 a.1 b.1 + a.2 b.2 + a.3 b.3.
2675 @cindex @code{simplify_indexed()}
2676 @subsection Simplifying indexed expressions
2678 In addition to the few automatic simplifications that GiNaC performs on
2679 indexed expressions (such as re-ordering the indices of symmetric tensors
2680 and calculating traces and convolutions of matrices and predefined tensors)
2684 ex ex::simplify_indexed();
2685 ex ex::simplify_indexed(const scalar_products & sp);
2688 that performs some more expensive operations:
2691 @item it checks the consistency of free indices in sums in the same way
2692 @code{get_free_indices()} does
2693 @item it tries to give dummy indices that appear in different terms of a sum
2694 the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
2695 @item it (symbolically) calculates all possible dummy index summations/contractions
2696 with the predefined tensors (this will be explained in more detail in the
2698 @item it detects contractions that vanish for symmetry reasons, for example
2699 the contraction of a symmetric and a totally antisymmetric tensor
2700 @item as a special case of dummy index summation, it can replace scalar products
2701 of two tensors with a user-defined value
2704 The last point is done with the help of the @code{scalar_products} class
2705 which is used to store scalar products with known values (this is not an
2706 arithmetic class, you just pass it to @code{simplify_indexed()}):
2710 symbol A("A"), B("B"), C("C"), i_sym("i");
2714 sp.add(A, B, 0); // A and B are orthogonal
2715 sp.add(A, C, 0); // A and C are orthogonal
2716 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
2718 e = indexed(A + B, i) * indexed(A + C, i);
2720 // -> (B+A).i*(A+C).i
2722 cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
2728 The @code{scalar_products} object @code{sp} acts as a storage for the
2729 scalar products added to it with the @code{.add()} method. This method
2730 takes three arguments: the two expressions of which the scalar product is
2731 taken, and the expression to replace it with.
2733 @cindex @code{expand()}
2734 The example above also illustrates a feature of the @code{expand()} method:
2735 if passed the @code{expand_indexed} option it will distribute indices
2736 over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
2738 @cindex @code{tensor} (class)
2739 @subsection Predefined tensors
2741 Some frequently used special tensors such as the delta, epsilon and metric
2742 tensors are predefined in GiNaC. They have special properties when
2743 contracted with other tensor expressions and some of them have constant
2744 matrix representations (they will evaluate to a number when numeric
2745 indices are specified).
2747 @cindex @code{delta_tensor()}
2748 @subsubsection Delta tensor
2750 The delta tensor takes two indices, is symmetric and has the matrix
2751 representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
2752 @code{delta_tensor()}:
2756 symbol A("A"), B("B");
2758 idx i(symbol("i"), 3), j(symbol("j"), 3),
2759 k(symbol("k"), 3), l(symbol("l"), 3);
2761 ex e = indexed(A, i, j) * indexed(B, k, l)
2762 * delta_tensor(i, k) * delta_tensor(j, l);
2763 cout << e.simplify_indexed() << endl;
2766 cout << delta_tensor(i, i) << endl;
2771 @cindex @code{metric_tensor()}
2772 @subsubsection General metric tensor
2774 The function @code{metric_tensor()} creates a general symmetric metric
2775 tensor with two indices that can be used to raise/lower tensor indices. The
2776 metric tensor is denoted as @samp{g} in the output and if its indices are of
2777 mixed variance it is automatically replaced by a delta tensor:
2783 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
2785 ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
2786 cout << e.simplify_indexed() << endl;
2789 e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
2790 cout << e.simplify_indexed() << endl;
2793 e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
2794 * metric_tensor(nu, rho);
2795 cout << e.simplify_indexed() << endl;
2798 e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
2799 * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
2800 + indexed(A, mu.toggle_variance(), rho));
2801 cout << e.simplify_indexed() << endl;
2806 @cindex @code{lorentz_g()}
2807 @subsubsection Minkowski metric tensor
2809 The Minkowski metric tensor is a special metric tensor with a constant
2810 matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
2811 signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
2812 It is created with the function @code{lorentz_g()} (although it is output as
2817 varidx mu(symbol("mu"), 4);
2819 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2820 * lorentz_g(mu, varidx(0, 4)); // negative signature
2821 cout << e.simplify_indexed() << endl;
2824 e = delta_tensor(varidx(0, 4), mu.toggle_variance())
2825 * lorentz_g(mu, varidx(0, 4), true); // positive signature
2826 cout << e.simplify_indexed() << endl;
2831 @cindex @code{spinor_metric()}
2832 @subsubsection Spinor metric tensor
2834 The function @code{spinor_metric()} creates an antisymmetric tensor with
2835 two indices that is used to raise/lower indices of 2-component spinors.
2836 It is output as @samp{eps}:
2842 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
2843 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
2845 e = spinor_metric(A, B) * indexed(psi, B_co);
2846 cout << e.simplify_indexed() << endl;
2849 e = spinor_metric(A, B) * indexed(psi, A_co);
2850 cout << e.simplify_indexed() << endl;
2853 e = spinor_metric(A_co, B_co) * indexed(psi, B);
2854 cout << e.simplify_indexed() << endl;
2857 e = spinor_metric(A_co, B_co) * indexed(psi, A);
2858 cout << e.simplify_indexed() << endl;
2861 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
2862 cout << e.simplify_indexed() << endl;
2865 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
2866 cout << e.simplify_indexed() << endl;
2871 The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
2873 @cindex @code{epsilon_tensor()}
2874 @cindex @code{lorentz_eps()}
2875 @subsubsection Epsilon tensor
2877 The epsilon tensor is totally antisymmetric, its number of indices is equal
2878 to the dimension of the index space (the indices must all be of the same
2879 numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
2880 defined to be 1. Its behavior with indices that have a variance also
2881 depends on the signature of the metric. Epsilon tensors are output as
2884 There are three functions defined to create epsilon tensors in 2, 3 and 4
2888 ex epsilon_tensor(const ex & i1, const ex & i2);
2889 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
2890 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
2891 bool pos_sig = false);
2894 The first two functions create an epsilon tensor in 2 or 3 Euclidean
2895 dimensions, the last function creates an epsilon tensor in a 4-dimensional
2896 Minkowski space (the last @code{bool} argument specifies whether the metric
2897 has negative or positive signature, as in the case of the Minkowski metric
2902 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
2903 sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
2904 e = lorentz_eps(mu, nu, rho, sig) *
2905 lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
2906 cout << simplify_indexed(e) << endl;
2907 // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
2909 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
2910 symbol A("A"), B("B");
2911 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
2912 cout << simplify_indexed(e) << endl;
2913 // -> -B.k*A.j*eps.i.k.j
2914 e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
2915 cout << simplify_indexed(e) << endl;
2920 @subsection Linear algebra
2922 The @code{matrix} class can be used with indices to do some simple linear
2923 algebra (linear combinations and products of vectors and matrices, traces
2924 and scalar products):
2928 idx i(symbol("i"), 2), j(symbol("j"), 2);
2929 symbol x("x"), y("y");
2931 // A is a 2x2 matrix, X is a 2x1 vector
2932 matrix A(2, 2), X(2, 1);
2937 cout << indexed(A, i, i) << endl;
2940 ex e = indexed(A, i, j) * indexed(X, j);
2941 cout << e.simplify_indexed() << endl;
2942 // -> [[2*y+x],[4*y+3*x]].i
2944 e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
2945 cout << e.simplify_indexed() << endl;
2946 // -> [[3*y+3*x,6*y+2*x]].j
2950 You can of course obtain the same results with the @code{matrix::add()},
2951 @code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
2952 but with indices you don't have to worry about transposing matrices.
2954 Matrix indices always start at 0 and their dimension must match the number
2955 of rows/columns of the matrix. Matrices with one row or one column are
2956 vectors and can have one or two indices (it doesn't matter whether it's a
2957 row or a column vector). Other matrices must have two indices.
2959 You should be careful when using indices with variance on matrices. GiNaC
2960 doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
2961 @samp{F.mu.nu} are different matrices. In this case you should use only
2962 one form for @samp{F} and explicitly multiply it with a matrix representation
2963 of the metric tensor.
2966 @node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
2967 @c node-name, next, previous, up
2968 @section Non-commutative objects
2970 GiNaC is equipped to handle certain non-commutative algebras. Three classes of
2971 non-commutative objects are built-in which are mostly of use in high energy
2975 @item Clifford (Dirac) algebra (class @code{clifford})
2976 @item su(3) Lie algebra (class @code{color})
2977 @item Matrices (unindexed) (class @code{matrix})
2980 The @code{clifford} and @code{color} classes are subclasses of
2981 @code{indexed} because the elements of these algebras usually carry
2982 indices. The @code{matrix} class is described in more detail in
2985 Unlike most computer algebra systems, GiNaC does not primarily provide an
2986 operator (often denoted @samp{&*}) for representing inert products of
2987 arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
2988 classes of objects involved, and non-commutative products are formed with
2989 the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
2990 figuring out by itself which objects commutate and will group the factors
2991 by their class. Consider this example:
2995 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
2996 idx a(symbol("a"), 8), b(symbol("b"), 8);
2997 ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
2999 // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
3003 As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
3004 groups the non-commutative factors (the gammas and the su(3) generators)
3005 together while preserving the order of factors within each class (because
3006 Clifford objects commutate with color objects). The resulting expression is a
3007 @emph{commutative} product with two factors that are themselves non-commutative
3008 products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
3009 parentheses are placed around the non-commutative products in the output.
3011 @cindex @code{ncmul} (class)
3012 Non-commutative products are internally represented by objects of the class
3013 @code{ncmul}, as opposed to commutative products which are handled by the
3014 @code{mul} class. You will normally not have to worry about this distinction,
3017 The advantage of this approach is that you never have to worry about using
3018 (or forgetting to use) a special operator when constructing non-commutative
3019 expressions. Also, non-commutative products in GiNaC are more intelligent
3020 than in other computer algebra systems; they can, for example, automatically
3021 canonicalize themselves according to rules specified in the implementation
3022 of the non-commutative classes. The drawback is that to work with other than
3023 the built-in algebras you have to implement new classes yourself. Both
3024 symbols and user-defined functions can be specified as being non-commutative.
3026 @cindex @code{return_type()}
3027 @cindex @code{return_type_tinfo()}
3028 Information about the commutativity of an object or expression can be
3029 obtained with the two member functions
3032 unsigned ex::return_type() const;
3033 unsigned ex::return_type_tinfo() const;
3036 The @code{return_type()} function returns one of three values (defined in
3037 the header file @file{flags.h}), corresponding to three categories of
3038 expressions in GiNaC:
3041 @item @code{return_types::commutative}: Commutates with everything. Most GiNaC
3042 classes are of this kind.
3043 @item @code{return_types::noncommutative}: Non-commutative, belonging to a
3044 certain class of non-commutative objects which can be determined with the
3045 @code{return_type_tinfo()} method. Expressions of this category commutate
3046 with everything except @code{noncommutative} expressions of the same
3048 @item @code{return_types::noncommutative_composite}: Non-commutative, composed
3049 of non-commutative objects of different classes. Expressions of this
3050 category don't commutate with any other @code{noncommutative} or
3051 @code{noncommutative_composite} expressions.
3054 The value returned by the @code{return_type_tinfo()} method is valid only
3055 when the return type of the expression is @code{noncommutative}. It is a
3056 value that is unique to the class of the object and usually one of the
3057 constants in @file{tinfos.h}, or derived therefrom.
3059 Here are a couple of examples:
3062 @multitable @columnfractions 0.33 0.33 0.34
3063 @item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
3064 @item @code{42} @tab @code{commutative} @tab -
3065 @item @code{2*x-y} @tab @code{commutative} @tab -
3066 @item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3067 @item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
3068 @item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
3069 @item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
3073 Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
3074 @code{TINFO_clifford} for objects with a representation label of zero.
3075 Other representation labels yield a different @code{return_type_tinfo()},
3076 but it's the same for any two objects with the same label. This is also true
3079 A last note: With the exception of matrices, positive integer powers of
3080 non-commutative objects are automatically expanded in GiNaC. For example,
3081 @code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
3082 non-commutative expressions).
3085 @cindex @code{clifford} (class)
3086 @subsection Clifford algebra
3089 Clifford algebras are supported in two flavours: Dirac gamma
3090 matrices (more physical) and generic Clifford algebras (more
3093 @cindex @code{dirac_gamma()}
3094 @subsubsection Dirac gamma matrices
3095 Dirac gamma matrices (note that GiNaC doesn't treat them
3096 as matrices) are designated as @samp{gamma~mu} and satisfy
3097 @samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
3098 @samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
3099 constructed by the function
3102 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
3105 which takes two arguments: the index and a @dfn{representation label} in the
3106 range 0 to 255 which is used to distinguish elements of different Clifford
3107 algebras (this is also called a @dfn{spin line index}). Gammas with different
3108 labels commutate with each other. The dimension of the index can be 4 or (in
3109 the framework of dimensional regularization) any symbolic value. Spinor
3110 indices on Dirac gammas are not supported in GiNaC.
3112 @cindex @code{dirac_ONE()}
3113 The unity element of a Clifford algebra is constructed by
3116 ex dirac_ONE(unsigned char rl = 0);
3119 @strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
3120 multiples of the unity element, even though it's customary to omit it.
3121 E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
3122 write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
3123 GiNaC will complain and/or produce incorrect results.
3125 @cindex @code{dirac_gamma5()}
3126 There is a special element @samp{gamma5} that commutates with all other
3127 gammas, has a unit square, and in 4 dimensions equals
3128 @samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
3131 ex dirac_gamma5(unsigned char rl = 0);
3134 @cindex @code{dirac_gammaL()}
3135 @cindex @code{dirac_gammaR()}
3136 The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
3137 objects, constructed by
3140 ex dirac_gammaL(unsigned char rl = 0);
3141 ex dirac_gammaR(unsigned char rl = 0);
3144 They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
3145 and @samp{gammaL gammaR = gammaR gammaL = 0}.
3147 @cindex @code{dirac_slash()}
3148 Finally, the function
3151 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
3154 creates a term that represents a contraction of @samp{e} with the Dirac
3155 Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
3156 with a unique index whose dimension is given by the @code{dim} argument).
3157 Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
3159 In products of dirac gammas, superfluous unity elements are automatically
3160 removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
3161 and @samp{gammaR} are moved to the front.
3163 The @code{simplify_indexed()} function performs contractions in gamma strings,
3169 symbol a("a"), b("b"), D("D");
3170 varidx mu(symbol("mu"), D);
3171 ex e = dirac_gamma(mu) * dirac_slash(a, D)
3172 * dirac_gamma(mu.toggle_variance());
3174 // -> gamma~mu*a\*gamma.mu
3175 e = e.simplify_indexed();
3178 cout << e.subs(D == 4) << endl;
3184 @cindex @code{dirac_trace()}
3185 To calculate the trace of an expression containing strings of Dirac gammas
3186 you use one of the functions
3189 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
3190 const ex & trONE = 4);
3191 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
3192 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
3195 These functions take the trace over all gammas in the specified set @code{rls}
3196 or list @code{rll} of representation labels, or the single label @code{rl};
3197 gammas with other labels are left standing. The last argument to
3198 @code{dirac_trace()} is the value to be returned for the trace of the unity
3199 element, which defaults to 4.
3201 The @code{dirac_trace()} function is a linear functional that is equal to the
3202 ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
3203 functional is not cyclic in
3206 dimensions when acting on
3207 expressions containing @samp{gamma5}, so it's not a proper trace. This
3208 @samp{gamma5} scheme is described in greater detail in
3209 @cite{The Role of gamma5 in Dimensional Regularization}.
3211 The value of the trace itself is also usually different in 4 and in
3219 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
3220 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3221 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3222 cout << dirac_trace(e).simplify_indexed() << endl;
3229 varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
3230 ex e = dirac_gamma(mu) * dirac_gamma(nu) *
3231 dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
3232 cout << dirac_trace(e).simplify_indexed() << endl;
3233 // -> 8*eta~rho~nu-4*eta~rho~nu*D
3237 Here is an example for using @code{dirac_trace()} to compute a value that
3238 appears in the calculation of the one-loop vacuum polarization amplitude in
3243 symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
3244 varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
3247 sp.add(l, l, pow(l, 2));
3248 sp.add(l, q, ldotq);
3250 ex e = dirac_gamma(mu) *
3251 (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
3252 dirac_gamma(mu.toggle_variance()) *
3253 (dirac_slash(l, D) + m * dirac_ONE());
3254 e = dirac_trace(e).simplify_indexed(sp);
3255 e = e.collect(lst(l, ldotq, m));
3257 // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
3261 The @code{canonicalize_clifford()} function reorders all gamma products that
3262 appear in an expression to a canonical (but not necessarily simple) form.
3263 You can use this to compare two expressions or for further simplifications:
3267 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
3268 ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
3270 // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
3272 e = canonicalize_clifford(e);
3274 // -> 2*ONE*eta~mu~nu
3278 @cindex @code{clifford_unit()}
3279 @subsubsection A generic Clifford algebra
3281 A generic Clifford algebra, i.e. a
3285 dimensional algebra with
3289 satisfying the identities
3291 $e_i e_j + e_j e_i = M(i, j) + M(j, i) $
3294 e~i e~j + e~j e~i = M(i, j) + M(j, i)
3296 for some bilinear form (@code{metric})
3297 @math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
3298 and contain symbolic entries. Such generators are created by the
3302 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
3303 bool anticommuting = false);
3306 where @code{mu} should be a @code{varidx} class object indexing the
3307 generators, an index @code{mu} with a numeric value may be of type
3309 Parameter @code{metr} defines the metric @math{M(i, j)} and can be
3310 represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
3311 object. In fact, any expression either with two free indices or without
3312 indices at all is admitted as @code{metr}. In the later case an @code{indexed}
3313 object with two newly created indices with @code{metr} as its
3314 @code{op(0)} will be used.
3315 Optional parameter @code{rl} allows to distinguish different
3316 Clifford algebras, which will commute with each other. The last
3317 optional parameter @code{anticommuting} defines if the anticommuting
3320 $e_i e_j + e_j e_i = 0$)
3323 e~i e~j + e~j e~i = 0)
3325 will be used for contraction of Clifford units. If the @code{metric} is
3326 supplied by a @code{matrix} object, then the value of
3327 @code{anticommuting} is calculated automatically and the supplied one
3328 will be ignored. One can overcome this by giving @code{metric} through
3329 matrix wrapped into an @code{indexed} object.
3331 Note that the call @code{clifford_unit(mu, minkmetric())} creates
3332 something very close to @code{dirac_gamma(mu)}, although
3333 @code{dirac_gamma} have more efficient simplification mechanism.
3334 @cindex @code{clifford::get_metric()}
3335 The method @code{clifford::get_metric()} returns a metric defining this
3337 @cindex @code{clifford::is_anticommuting()}
3338 The method @code{clifford::is_anticommuting()} returns the
3339 @code{anticommuting} property of a unit.
3341 If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
3342 the Clifford algebra units with a call like that
3345 ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
3348 since this may yield some further automatic simplifications. Again, for a
3349 metric defined through a @code{matrix} such a symmetry is detected
3352 Individual generators of a Clifford algebra can be accessed in several
3358 varidx nu(symbol("nu"), 4);
3360 ex M = diag_matrix(lst(1, -1, 0, s));
3361 ex e = clifford_unit(nu, M);
3362 ex e0 = e.subs(nu == 0);
3363 ex e1 = e.subs(nu == 1);
3364 ex e2 = e.subs(nu == 2);
3365 ex e3 = e.subs(nu == 3);
3370 will produce four anti-commuting generators of a Clifford algebra with properties
3372 $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
3375 @code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
3376 @code{pow(e3, 2) = s}.
3379 @cindex @code{lst_to_clifford()}
3380 A similar effect can be achieved from the function
3383 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
3384 unsigned char rl = 0, bool anticommuting = false);
3385 ex lst_to_clifford(const ex & v, const ex & e);
3388 which converts a list or vector
3390 $v = (v^0, v^1, ..., v^n)$
3393 @samp{v = (v~0, v~1, ..., v~n)}
3398 $v^0 e_0 + v^1 e_1 + ... + v^n e_n$
3401 @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
3404 directly supplied in the second form of the procedure. In the first form
3405 the Clifford unit @samp{e.k} is generated by the call of
3406 @code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
3407 with the help of @code{lst_to_clifford()} as follows
3412 varidx nu(symbol("nu"), 4);
3414 ex M = diag_matrix(lst(1, -1, 0, s));
3415 ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
3416 ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
3417 ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
3418 ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
3423 @cindex @code{clifford_to_lst()}
3424 There is the inverse function
3427 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
3430 which takes an expression @code{e} and tries to find a list
3432 $v = (v^0, v^1, ..., v^n)$
3435 @samp{v = (v~0, v~1, ..., v~n)}
3439 $e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
3442 @samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
3444 with respect to the given Clifford units @code{c} and with none of the
3445 @samp{v~k} containing Clifford units @code{c} (of course, this
3446 may be impossible). This function can use an @code{algebraic} method
3447 (default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
3449 $(e c_k + c_k e)/c_k^2$. If $c_k^2$
3452 @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
3454 is zero or is not @code{numeric} for some @samp{k}
3455 then the method will be automatically changed to symbolic. The same effect
3456 is obtained by the assignment (@code{algebraic = false}) in the procedure call.
3458 @cindex @code{clifford_prime()}
3459 @cindex @code{clifford_star()}
3460 @cindex @code{clifford_bar()}
3461 There are several functions for (anti-)automorphisms of Clifford algebras:
3464 ex clifford_prime(const ex & e)
3465 inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
3466 inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
3469 The automorphism of a Clifford algebra @code{clifford_prime()} simply
3470 changes signs of all Clifford units in the expression. The reversion
3471 of a Clifford algebra @code{clifford_star()} coincides with the
3472 @code{conjugate()} method and effectively reverses the order of Clifford
3473 units in any product. Finally the main anti-automorphism
3474 of a Clifford algebra @code{clifford_bar()} is the composition of the
3475 previous two, i.e. it makes the reversion and changes signs of all Clifford units
3476 in a product. These functions correspond to the notations
3491 used in Clifford algebra textbooks.
3493 @cindex @code{clifford_norm()}
3497 ex clifford_norm(const ex & e);
3500 @cindex @code{clifford_inverse()}
3501 calculates the norm of a Clifford number from the expression
3503 $||e||^2 = e\overline{e}$.
3506 @code{||e||^2 = e \bar@{e@}}
3508 The inverse of a Clifford expression is returned by the function
3511 ex clifford_inverse(const ex & e);
3514 which calculates it as
3516 $e^{-1} = \overline{e}/||e||^2$.
3519 @math{e^@{-1@} = \bar@{e@}/||e||^2}
3528 then an exception is raised.
3530 @cindex @code{remove_dirac_ONE()}
3531 If a Clifford number happens to be a factor of
3532 @code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
3533 expression by the function
3536 ex remove_dirac_ONE(const ex & e);
3539 @cindex @code{canonicalize_clifford()}
3540 The function @code{canonicalize_clifford()} works for a
3541 generic Clifford algebra in a similar way as for Dirac gammas.
3543 The next provided function is
3545 @cindex @code{clifford_moebius_map()}
3547 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
3548 const ex & d, const ex & v, const ex & G,
3549 unsigned char rl = 0, bool anticommuting = false);
3550 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
3551 unsigned char rl = 0, bool anticommuting = false);
3554 It takes a list or vector @code{v} and makes the Moebius (conformal or
3555 linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
3556 the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
3557 the metric of the surrounding (pseudo-)Euclidean space. This can be an
3558 indexed object, tensormetric, matrix or a Clifford unit, in the later
3559 case the optional parameters @code{rl} and @code{anticommuting} are
3560 ignored even if supplied. Depending from the type of @code{v} the
3561 returned value of this function is either a vector or a list holding vector's
3564 @cindex @code{clifford_max_label()}
3565 Finally the function
3568 char clifford_max_label(const ex & e, bool ignore_ONE = false);
3571 can detect a presence of Clifford objects in the expression @code{e}: if
3572 such objects are found it returns the maximal
3573 @code{representation_label} of them, otherwise @code{-1}. The optional
3574 parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
3575 be ignored during the search.
3577 LaTeX output for Clifford units looks like
3578 @code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
3579 @code{representation_label} and @code{\nu} is the index of the
3580 corresponding unit. This provides a flexible typesetting with a suitable
3581 defintion of the @code{\clifford} command. For example, the definition
3583 \newcommand@{\clifford@}[1][]@{@}
3585 typesets all Clifford units identically, while the alternative definition
3587 \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
3589 prints units with @code{representation_label=0} as
3596 with @code{representation_label=1} as
3603 and with @code{representation_label=2} as
3611 @cindex @code{color} (class)
3612 @subsection Color algebra
3614 @cindex @code{color_T()}
3615 For computations in quantum chromodynamics, GiNaC implements the base elements
3616 and structure constants of the su(3) Lie algebra (color algebra). The base
3617 elements @math{T_a} are constructed by the function
3620 ex color_T(const ex & a, unsigned char rl = 0);
3623 which takes two arguments: the index and a @dfn{representation label} in the
3624 range 0 to 255 which is used to distinguish elements of different color
3625 algebras. Objects with different labels commutate with each other. The
3626 dimension of the index must be exactly 8 and it should be of class @code{idx},
3629 @cindex @code{color_ONE()}
3630 The unity element of a color algebra is constructed by
3633 ex color_ONE(unsigned char rl = 0);
3636 @strong{Please notice:} You must always use @code{color_ONE()} when referring to
3637 multiples of the unity element, even though it's customary to omit it.
3638 E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
3639 write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
3640 GiNaC may produce incorrect results.
3642 @cindex @code{color_d()}
3643 @cindex @code{color_f()}
3647 ex color_d(const ex & a, const ex & b, const ex & c);
3648 ex color_f(const ex & a, const ex & b, const ex & c);
3651 create the symmetric and antisymmetric structure constants @math{d_abc} and
3652 @math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
3653 and @math{[T_a, T_b] = i f_abc T_c}.
3655 These functions evaluate to their numerical values,
3656 if you supply numeric indices to them. The index values should be in
3657 the range from 1 to 8, not from 0 to 7. This departure from usual conventions
3658 goes along better with the notations used in physical literature.
3660 @cindex @code{color_h()}
3661 There's an additional function
3664 ex color_h(const ex & a, const ex & b, const ex & c);
3667 which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
3669 The function @code{simplify_indexed()} performs some simplifications on
3670 expressions containing color objects:
3675 idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
3676 k(symbol("k"), 8), l(symbol("l"), 8);
3678 e = color_d(a, b, l) * color_f(a, b, k);
3679 cout << e.simplify_indexed() << endl;
3682 e = color_d(a, b, l) * color_d(a, b, k);
3683 cout << e.simplify_indexed() << endl;
3686 e = color_f(l, a, b) * color_f(a, b, k);
3687 cout << e.simplify_indexed() << endl;
3690 e = color_h(a, b, c) * color_h(a, b, c);
3691 cout << e.simplify_indexed() << endl;
3694 e = color_h(a, b, c) * color_T(b) * color_T(c);
3695 cout << e.simplify_indexed() << endl;
3698 e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
3699 cout << e.simplify_indexed() << endl;
3702 e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
3703 cout << e.simplify_indexed() << endl;
3704 // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
3708 @cindex @code{color_trace()}
3709 To calculate the trace of an expression containing color objects you use one
3713 ex color_trace(const ex & e, const std::set<unsigned char> & rls);
3714 ex color_trace(const ex & e, const lst & rll);
3715 ex color_trace(const ex & e, unsigned char rl = 0);
3718 These functions take the trace over all color @samp{T} objects in the
3719 specified set @code{rls} or list @code{rll} of representation labels, or the
3720 single label @code{rl}; @samp{T}s with other labels are left standing. For
3725 e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
3727 // -> -I*f.a.c.b+d.a.c.b
3732 @node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
3733 @c node-name, next, previous, up
3736 @cindex @code{exhashmap} (class)
3738 For your convenience, GiNaC offers the container template @code{exhashmap<T>}
3739 that can be used as a drop-in replacement for the STL
3740 @code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
3741 typically constant-time, element look-up than @code{map<>}.
3743 @code{exhashmap<>} supports all @code{map<>} members and operations, with the
3744 following differences:
3748 no @code{lower_bound()} and @code{upper_bound()} methods
3750 no reverse iterators, no @code{rbegin()}/@code{rend()}
3752 no @code{operator<(exhashmap, exhashmap)}
3754 the comparison function object @code{key_compare} is hardcoded to
3757 the constructor @code{exhashmap(size_t n)} allows specifying the minimum
3758 initial hash table size (the actual table size after construction may be
3759 larger than the specified value)
3761 the method @code{size_t bucket_count()} returns the current size of the hash
3764 @code{insert()} and @code{erase()} operations invalidate all iterators
3768 @node Methods and functions, Information about expressions, Hash maps, Top
3769 @c node-name, next, previous, up
3770 @chapter Methods and functions
3773 In this chapter the most important algorithms provided by GiNaC will be
3774 described. Some of them are implemented as functions on expressions,
3775 others are implemented as methods provided by expression objects. If
3776 they are methods, there exists a wrapper function around it, so you can
3777 alternatively call it in a functional way as shown in the simple
3782 cout << "As method: " << sin(1).evalf() << endl;
3783 cout << "As function: " << evalf(sin(1)) << endl;
3787 @cindex @code{subs()}
3788 The general rule is that wherever methods accept one or more parameters
3789 (@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
3790 wrapper accepts is the same but preceded by the object to act on
3791 (@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
3792 most natural one in an OO model but it may lead to confusion for MapleV
3793 users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
3794 would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
3795 @code{A} and @code{x}). On the other hand, since MapleV returns 3 on
3796 @code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
3797 coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
3798 here. Also, users of MuPAD will in most cases feel more comfortable
3799 with GiNaC's convention. All function wrappers are implemented
3800 as simple inline functions which just call the corresponding method and
3801 are only provided for users uncomfortable with OO who are dead set to
3802 avoid method invocations. Generally, nested function wrappers are much
3803 harder to read than a sequence of methods and should therefore be
3804 avoided if possible. On the other hand, not everything in GiNaC is a
3805 method on class @code{ex} and sometimes calling a function cannot be
3809 * Information about expressions::
3810 * Numerical evaluation::
3811 * Substituting expressions::
3812 * Pattern matching and advanced substitutions::
3813 * Applying a function on subexpressions::
3814 * Visitors and tree traversal::
3815 * Polynomial arithmetic:: Working with polynomials.
3816 * Rational expressions:: Working with rational functions.
3817 * Symbolic differentiation::
3818 * Series expansion:: Taylor and Laurent expansion.
3820 * Built-in functions:: List of predefined mathematical functions.
3821 * Multiple polylogarithms::
3822 * Complex expressions::
3823 * Solving linear systems of equations::
3824 * Input/output:: Input and output of expressions.
3828 @node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
3829 @c node-name, next, previous, up
3830 @section Getting information about expressions
3832 @subsection Checking expression types
3833 @cindex @code{is_a<@dots{}>()}
3834 @cindex @code{is_exactly_a<@dots{}>()}
3835 @cindex @code{ex_to<@dots{}>()}
3836 @cindex Converting @code{ex} to other classes
3837 @cindex @code{info()}
3838 @cindex @code{return_type()}
3839 @cindex @code{return_type_tinfo()}
3841 Sometimes it's useful to check whether a given expression is a plain number,
3842 a sum, a polynomial with integer coefficients, or of some other specific type.
3843 GiNaC provides a couple of functions for this:
3846 bool is_a<T>(const ex & e);
3847 bool is_exactly_a<T>(const ex & e);
3848 bool ex::info(unsigned flag);
3849 unsigned ex::return_type() const;
3850 unsigned ex::return_type_tinfo() const;
3853 When the test made by @code{is_a<T>()} returns true, it is safe to call
3854 one of the functions @code{ex_to<T>()}, where @code{T} is one of the
3855 class names (@xref{The class hierarchy}, for a list of all classes). For
3856 example, assuming @code{e} is an @code{ex}:
3861 if (is_a<numeric>(e))
3862 numeric n = ex_to<numeric>(e);
3867 @code{is_a<T>(e)} allows you to check whether the top-level object of
3868 an expression @samp{e} is an instance of the GiNaC class @samp{T}
3869 (@xref{The class hierarchy}, for a list of all classes). This is most useful,
3870 e.g., for checking whether an expression is a number, a sum, or a product:
3877 is_a<numeric>(e1); // true
3878 is_a<numeric>(e2); // false
3879 is_a<add>(e1); // false
3880 is_a<add>(e2); // true
3881 is_a<mul>(e1); // false
3882 is_a<mul>(e2); // false
3886 In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
3887 top-level object of an expression @samp{e} is an instance of the GiNaC
3888 class @samp{T}, not including parent classes.
3890 The @code{info()} method is used for checking certain attributes of
3891 expressions. The possible values for the @code{flag} argument are defined
3892 in @file{ginac/flags.h}, the most important being explained in the following
3896 @multitable @columnfractions .30 .70
3897 @item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
3898 @item @code{numeric}
3899 @tab @dots{}a number (same as @code{is_a<numeric>(...)})
3901 @tab @dots{}a real number, symbol or constant (i.e. is not complex)
3902 @item @code{rational}
3903 @tab @dots{}an exact rational number (integers are rational, too)
3904 @item @code{integer}
3905 @tab @dots{}a (non-complex) integer
3906 @item @code{crational}
3907 @tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
3908 @item @code{cinteger}
3909 @tab @dots{}a (complex) integer (such as @math{2-3*I})
3910 @item @code{positive}
3911 @tab @dots{}not complex and greater than 0
3912 @item @code{negative}
3913 @tab @dots{}not complex and less than 0
3914 @item @code{nonnegative}
3915 @tab @dots{}not complex and greater than or equal to 0
3917 @tab @dots{}an integer greater than 0
3919 @tab @dots{}an integer less than 0
3920 @item @code{nonnegint}
3921 @tab @dots{}an integer greater than or equal to 0
3923 @tab @dots{}an even integer
3925 @tab @dots{}an odd integer
3927 @tab @dots{}a prime integer (probabilistic primality test)
3928 @item @code{relation}
3929 @tab @dots{}a relation (same as @code{is_a<relational>(...)})
3930 @item @code{relation_equal}
3931 @tab @dots{}a @code{==} relation
3932 @item @code{relation_not_equal}
3933 @tab @dots{}a @code{!=} relation
3934 @item @code{relation_less}
3935 @tab @dots{}a @code{<} relation
3936 @item @code{relation_less_or_equal}
3937 @tab @dots{}a @code{<=} relation
3938 @item @code{relation_greater}
3939 @tab @dots{}a @code{>} relation
3940 @item @code{relation_greater_or_equal}
3941 @tab @dots{}a @code{>=} relation
3943 @tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
3945 @tab @dots{}a list (same as @code{is_a<lst>(...)})
3946 @item @code{polynomial}
3947 @tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
3948 @item @code{integer_polynomial}
3949 @tab @dots{}a polynomial with (non-complex) integer coefficients
3950 @item @code{cinteger_polynomial}
3951 @tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
3952 @item @code{rational_polynomial}
3953 @tab @dots{}a polynomial with (non-complex) rational coefficients
3954 @item @code{crational_polynomial}
3955 @tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
3956 @item @code{rational_function}
3957 @tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
3958 @item @code{algebraic}
3959 @tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
3963 To determine whether an expression is commutative or non-commutative and if
3964 so, with which other expressions it would commutate, you use the methods
3965 @code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
3966 for an explanation of these.
3969 @subsection Accessing subexpressions
3972 Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
3973 @code{function}, act as containers for subexpressions. For example, the
3974 subexpressions of a sum (an @code{add} object) are the individual terms,
3975 and the subexpressions of a @code{function} are the function's arguments.
3977 @cindex @code{nops()}
3979 GiNaC provides several ways of accessing subexpressions. The first way is to
3984 ex ex::op(size_t i);
3987 @code{nops()} determines the number of subexpressions (operands) contained
3988 in the expression, while @code{op(i)} returns the @code{i}-th
3989 (0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
3990 @code{op(0)} will return the basis and @code{op(1)} the exponent. For
3991 @code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
3992 @math{i>0} are the indices.
3995 @cindex @code{const_iterator}
3996 The second way to access subexpressions is via the STL-style random-access
3997 iterator class @code{const_iterator} and the methods
4000 const_iterator ex::begin();
4001 const_iterator ex::end();
4004 @code{begin()} returns an iterator referring to the first subexpression;
4005 @code{end()} returns an iterator which is one-past the last subexpression.
4006 If the expression has no subexpressions, then @code{begin() == end()}. These
4007 iterators can also be used in conjunction with non-modifying STL algorithms.
4009 Here is an example that (non-recursively) prints the subexpressions of a
4010 given expression in three different ways:
4017 for (size_t i = 0; i != e.nops(); ++i)
4018 cout << e.op(i) << endl;
4021 for (const_iterator i = e.begin(); i != e.end(); ++i)
4024 // with iterators and STL copy()
4025 std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
4029 @cindex @code{const_preorder_iterator}
4030 @cindex @code{const_postorder_iterator}
4031 @code{op()}/@code{nops()} and @code{const_iterator} only access an
4032 expression's immediate children. GiNaC provides two additional iterator
4033 classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
4034 that iterate over all objects in an expression tree, in preorder or postorder,
4035 respectively. They are STL-style forward iterators, and are created with the
4039 const_preorder_iterator ex::preorder_begin();
4040 const_preorder_iterator ex::preorder_end();
4041 const_postorder_iterator ex::postorder_begin();
4042 const_postorder_iterator ex::postorder_end();
4045 The following example illustrates the differences between
4046 @code{const_iterator}, @code{const_preorder_iterator}, and
4047 @code{const_postorder_iterator}:
4051 symbol A("A"), B("B"), C("C");
4052 ex e = lst(lst(A, B), C);
4054 std::copy(e.begin(), e.end(),
4055 std::ostream_iterator<ex>(cout, "\n"));
4059 std::copy(e.preorder_begin(), e.preorder_end(),
4060 std::ostream_iterator<ex>(cout, "\n"));
4067 std::copy(e.postorder_begin(), e.postorder_end(),
4068 std::ostream_iterator<ex>(cout, "\n"));
4077 @cindex @code{relational} (class)
4078 Finally, the left-hand side and right-hand side expressions of objects of
4079 class @code{relational} (and only of these) can also be accessed with the
4088 @subsection Comparing expressions
4089 @cindex @code{is_equal()}
4090 @cindex @code{is_zero()}
4092 Expressions can be compared with the usual C++ relational operators like
4093 @code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
4094 the result is usually not determinable and the result will be @code{false},
4095 except in the case of the @code{!=} operator. You should also be aware that
4096 GiNaC will only do the most trivial test for equality (subtracting both
4097 expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
4100 Actually, if you construct an expression like @code{a == b}, this will be
4101 represented by an object of the @code{relational} class (@pxref{Relations})
4102 which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
4104 There are also two methods
4107 bool ex::is_equal(const ex & other);
4111 for checking whether one expression is equal to another, or equal to zero,
4112 respectively. See also the method @code{ex::is_zero_matrix()},
4116 @subsection Ordering expressions
4117 @cindex @code{ex_is_less} (class)
4118 @cindex @code{ex_is_equal} (class)
4119 @cindex @code{compare()}
4121 Sometimes it is necessary to establish a mathematically well-defined ordering
4122 on a set of arbitrary expressions, for example to use expressions as keys
4123 in a @code{std::map<>} container, or to bring a vector of expressions into
4124 a canonical order (which is done internally by GiNaC for sums and products).
4126 The operators @code{<}, @code{>} etc. described in the last section cannot
4127 be used for this, as they don't implement an ordering relation in the
4128 mathematical sense. In particular, they are not guaranteed to be
4129 antisymmetric: if @samp{a} and @samp{b} are different expressions, and
4130 @code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
4133 By default, STL classes and algorithms use the @code{<} and @code{==}
4134 operators to compare objects, which are unsuitable for expressions, but GiNaC
4135 provides two functors that can be supplied as proper binary comparison
4136 predicates to the STL:
4139 class ex_is_less : public std::binary_function<ex, ex, bool> @{
4141 bool operator()(const ex &lh, const ex &rh) const;
4144 class ex_is_equal : public std::binary_function<ex, ex, bool> @{
4146 bool operator()(const ex &lh, const ex &rh) const;
4150 For example, to define a @code{map} that maps expressions to strings you
4154 std::map<ex, std::string, ex_is_less> myMap;
4157 Omitting the @code{ex_is_less} template parameter will introduce spurious
4158 bugs because the map operates improperly.
4160 Other examples for the use of the functors:
4168 std::sort(v.begin(), v.end(), ex_is_less());
4170 // count the number of expressions equal to '1'
4171 unsigned num_ones = std::count_if(v.begin(), v.end(),
4172 std::bind2nd(ex_is_equal(), 1));
4175 The implementation of @code{ex_is_less} uses the member function
4178 int ex::compare(const ex & other) const;
4181 which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
4182 if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
4186 @node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
4187 @c node-name, next, previous, up
4188 @section Numerical evaluation
4189 @cindex @code{evalf()}
4191 GiNaC keeps algebraic expressions, numbers and constants in their exact form.
4192 To evaluate them using floating-point arithmetic you need to call
4195 ex ex::evalf(int level = 0) const;
4198 @cindex @code{Digits}
4199 The accuracy of the evaluation is controlled by the global object @code{Digits}
4200 which can be assigned an integer value. The default value of @code{Digits}
4201 is 17. @xref{Numbers}, for more information and examples.
4203 To evaluate an expression to a @code{double} floating-point number you can
4204 call @code{evalf()} followed by @code{numeric::to_double()}, like this:
4208 // Approximate sin(x/Pi)
4210 ex e = series(sin(x/Pi), x == 0, 6);
4212 // Evaluate numerically at x=0.1
4213 ex f = evalf(e.subs(x == 0.1));
4215 // ex_to<numeric> is an unsafe cast, so check the type first
4216 if (is_a<numeric>(f)) @{
4217 double d = ex_to<numeric>(f).to_double();
4226 @node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
4227 @c node-name, next, previous, up
4228 @section Substituting expressions
4229 @cindex @code{subs()}
4231 Algebraic objects inside expressions can be replaced with arbitrary
4232 expressions via the @code{.subs()} method:
4235 ex ex::subs(const ex & e, unsigned options = 0);
4236 ex ex::subs(const exmap & m, unsigned options = 0);
4237 ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
4240 In the first form, @code{subs()} accepts a relational of the form
4241 @samp{object == expression} or a @code{lst} of such relationals:
4245 symbol x("x"), y("y");
4247 ex e1 = 2*x^2-4*x+3;
4248 cout << "e1(7) = " << e1.subs(x == 7) << endl;
4252 cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
4257 If you specify multiple substitutions, they are performed in parallel, so e.g.
4258 @code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
4260 The second form of @code{subs()} takes an @code{exmap} object which is a
4261 pair associative container that maps expressions to expressions (currently
4262 implemented as a @code{std::map}). This is the most efficient one of the
4263 three @code{subs()} forms and should be used when the number of objects to
4264 be substituted is large or unknown.
4266 Using this form, the second example from above would look like this:
4270 symbol x("x"), y("y");
4276 cout << "e2(-2, 4) = " << e2.subs(m) << endl;
4280 The third form of @code{subs()} takes two lists, one for the objects to be
4281 replaced and one for the expressions to be substituted (both lists must
4282 contain the same number of elements). Using this form, you would write
4286 symbol x("x"), y("y");
4289 cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
4293 The optional last argument to @code{subs()} is a combination of
4294 @code{subs_options} flags. There are three options available:
4295 @code{subs_options::no_pattern} disables pattern matching, which makes
4296 large @code{subs()} operations significantly faster if you are not using
4297 patterns. The second option, @code{subs_options::algebraic} enables
4298 algebraic substitutions in products and powers.
4299 @ref{Pattern matching and advanced substitutions}, for more information
4300 about patterns and algebraic substitutions. The third option,
4301 @code{subs_options::no_index_renaming} disables the feature that dummy
4302 indices are renamed if the subsitution could give a result in which a
4303 dummy index occurs more than two times. This is sometimes necessary if
4304 you want to use @code{subs()} to rename your dummy indices.
4306 @code{subs()} performs syntactic substitution of any complete algebraic
4307 object; it does not try to match sub-expressions as is demonstrated by the
4312 symbol x("x"), y("y"), z("z");
4314 ex e1 = pow(x+y, 2);
4315 cout << e1.subs(x+y == 4) << endl;
4318 ex e2 = sin(x)*sin(y)*cos(x);
4319 cout << e2.subs(sin(x) == cos(x)) << endl;
4320 // -> cos(x)^2*sin(y)
4323 cout << e3.subs(x+y == 4) << endl;
4325 // (and not 4+z as one might expect)
4329 A more powerful form of substitution using wildcards is described in the
4333 @node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
4334 @c node-name, next, previous, up
4335 @section Pattern matching and advanced substitutions
4336 @cindex @code{wildcard} (class)
4337 @cindex Pattern matching
4339 GiNaC allows the use of patterns for checking whether an expression is of a
4340 certain form or contains subexpressions of a certain form, and for
4341 substituting expressions in a more general way.
4343 A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
4344 A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
4345 represents an arbitrary expression. Every wildcard has a @dfn{label} which is
4346 an unsigned integer number to allow having multiple different wildcards in a
4347 pattern. Wildcards are printed as @samp{$label} (this is also the way they
4348 are specified in @command{ginsh}). In C++ code, wildcard objects are created
4352 ex wild(unsigned label = 0);
4355 which is simply a wrapper for the @code{wildcard()} constructor with a shorter
4358 Some examples for patterns:
4360 @multitable @columnfractions .5 .5
4361 @item @strong{Constructed as} @tab @strong{Output as}
4362 @item @code{wild()} @tab @samp{$0}
4363 @item @code{pow(x,wild())} @tab @samp{x^$0}
4364 @item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
4365 @item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
4371 @item Wildcards behave like symbols and are subject to the same algebraic
4372 rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
4373 @item As shown in the last example, to use wildcards for indices you have to
4374 use them as the value of an @code{idx} object. This is because indices must
4375 always be of class @code{idx} (or a subclass).
4376 @item Wildcards only represent expressions or subexpressions. It is not
4377 possible to use them as placeholders for other properties like index
4378 dimension or variance, representation labels, symmetry of indexed objects
4380 @item Because wildcards are commutative, it is not possible to use wildcards
4381 as part of noncommutative products.
4382 @item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
4383 are also valid patterns.
4386 @subsection Matching expressions
4387 @cindex @code{match()}
4388 The most basic application of patterns is to check whether an expression
4389 matches a given pattern. This is done by the function
4392 bool ex::match(const ex & pattern);
4393 bool ex::match(const ex & pattern, lst & repls);
4396 This function returns @code{true} when the expression matches the pattern
4397 and @code{false} if it doesn't. If used in the second form, the actual
4398 subexpressions matched by the wildcards get returned in the @code{repls}
4399 object as a list of relations of the form @samp{wildcard == expression}.
4400 If @code{match()} returns false, the state of @code{repls} is undefined.
4401 For reproducible results, the list should be empty when passed to
4402 @code{match()}, but it is also possible to find similarities in multiple
4403 expressions by passing in the result of a previous match.
4405 The matching algorithm works as follows:
4408 @item A single wildcard matches any expression. If one wildcard appears
4409 multiple times in a pattern, it must match the same expression in all
4410 places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
4411 @samp{x*(x+1)} but not @samp{x*(y+1)}).
4412 @item If the expression is not of the same class as the pattern, the match
4413 fails (i.e. a sum only matches a sum, a function only matches a function,
4415 @item If the pattern is a function, it only matches the same function
4416 (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
4417 @item Except for sums and products, the match fails if the number of
4418 subexpressions (@code{nops()}) is not equal to the number of subexpressions
4420 @item If there are no subexpressions, the expressions and the pattern must
4421 be equal (in the sense of @code{is_equal()}).
4422 @item Except for sums and products, each subexpression (@code{op()}) must
4423 match the corresponding subexpression of the pattern.
4426 Sums (@code{add}) and products (@code{mul}) are treated in a special way to
4427 account for their commutativity and associativity:
4430 @item If the pattern contains a term or factor that is a single wildcard,
4431 this one is used as the @dfn{global wildcard}. If there is more than one
4432 such wildcard, one of them is chosen as the global wildcard in a random
4434 @item Every term/factor of the pattern, except the global wildcard, is
4435 matched against every term of the expression in sequence. If no match is
4436 found, the whole match fails. Terms that did match are not considered in
4438 @item If there are no unmatched terms left, the match succeeds. Otherwise
4439 the match fails unless there is a global wildcard in the pattern, in
4440 which case this wildcard matches the remaining terms.
4443 In general, having more than one single wildcard as a term of a sum or a
4444 factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
4447 Here are some examples in @command{ginsh} to demonstrate how it works (the
4448 @code{match()} function in @command{ginsh} returns @samp{FAIL} if the
4449 match fails, and the list of wildcard replacements otherwise):
4452 > match((x+y)^a,(x+y)^a);
4454 > match((x+y)^a,(x+y)^b);
4456 > match((x+y)^a,$1^$2);
4458 > match((x+y)^a,$1^$1);
4460 > match((x+y)^(x+y),$1^$1);
4462 > match((x+y)^(x+y),$1^$2);
4464 > match((a+b)*(a+c),($1+b)*($1+c));
4466 > match((a+b)*(a+c),(a+$1)*(a+$2));
4468 (Unpredictable. The result might also be [$1==c,$2==b].)
4469 > match((a+b)*(a+c),($1+$2)*($1+$3));
4470 (The result is undefined. Due to the sequential nature of the algorithm
4471 and the re-ordering of terms in GiNaC, the match for the first factor
4472 may be @{$1==a,$2==b@} in which case the match for the second factor
4473 succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
4475 > match(a*(x+y)+a*z+b,a*$1+$2);
4476 (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
4477 @{$1=x+y,$2=a*z+b@}.)
4478 > match(a+b+c+d+e+f,c);
4480 > match(a+b+c+d+e+f,c+$0);
4482 > match(a+b+c+d+e+f,c+e+$0);
4484 > match(a+b,a+b+$0);
4486 > match(a*b^2,a^$1*b^$2);
4488 (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
4489 even though a==a^1.)
4490 > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
4492 > match(atan2(y,x^2),atan2(y,$0));
4496 @subsection Matching parts of expressions
4497 @cindex @code{has()}
4498 A more general way to look for patterns in expressions is provided by the
4502 bool ex::has(const ex & pattern);
4505 This function checks whether a pattern is matched by an expression itself or
4506 by any of its subexpressions.
4508 Again some examples in @command{ginsh} for illustration (in @command{ginsh},
4509 @code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
4512 > has(x*sin(x+y+2*a),y);
4514 > has(x*sin(x+y+2*a),x+y);
4516 (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
4517 has the subexpressions "x", "y" and "2*a".)
4518 > has(x*sin(x+y+2*a),x+y+$1);
4520 (But this is possible.)
4521 > has(x*sin(2*(x+y)+2*a),x+y);
4523 (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
4524 which "x+y" is not a subexpression.)
4527 (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
4529 > has(4*x^2-x+3,$1*x);
4531 > has(4*x^2+x+3,$1*x);
4533 (Another possible pitfall. The first expression matches because the term
4534 "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
4535 contains a linear term you should use the coeff() function instead.)
4538 @cindex @code{find()}
4542 bool ex::find(const ex & pattern, lst & found);
4545 works a bit like @code{has()} but it doesn't stop upon finding the first
4546 match. Instead, it appends all found matches to the specified list. If there
4547 are multiple occurrences of the same expression, it is entered only once to
4548 the list. @code{find()} returns false if no matches were found (in
4549 @command{ginsh}, it returns an empty list):
4552 > find(1+x+x^2+x^3,x);
4554 > find(1+x+x^2+x^3,y);
4556 > find(1+x+x^2+x^3,x^$1);
4558 (Note the absence of "x".)
4559 > expand((sin(x)+sin(y))*(a+b));
4560 sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
4565 @subsection Substituting expressions
4566 @cindex @code{subs()}
4567 Probably the most useful application of patterns is to use them for
4568 substituting expressions with the @code{subs()} method. Wildcards can be
4569 used in the search patterns as well as in the replacement expressions, where
4570 they get replaced by the expressions matched by them. @code{subs()} doesn't
4571 know anything about algebra; it performs purely syntactic substitutions.
4576 > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
4578 > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
4580 > subs((a+b+c)^2,a+b==x);
4582 > subs((a+b+c)^2,a+b+$1==x+$1);
4584 > subs(a+2*b,a+b==x);
4586 > subs(4*x^3-2*x^2+5*x-1,x==a);
4588 > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
4590 > subs(sin(1+sin(x)),sin($1)==cos($1));
4592 > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
4596 The last example would be written in C++ in this way:
4600 symbol a("a"), b("b"), x("x"), y("y");
4601 e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
4602 e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
4603 cout << e.expand() << endl;
4608 @subsection The option algebraic
4609 Both @code{has()} and @code{subs()} take an optional argument to pass them
4610 extra options. This section describes what happens if you give the former
4611 the option @code{has_options::algebraic} or the latter
4612 @code{subs:options::algebraic}. In that case the matching condition for
4613 powers and multiplications is changed in such a way that they become
4614 more intuitive. Intuition says that @code{x*y} is a part of @code{x*y*z}.
4615 If you use these options you will find that
4616 @code{(x*y*z).has(x*y, has_options::algebraic)} indeed returns true.
4617 Besides matching some of the factors of a product also powers match as
4618 often as is possible without getting negative exponents. For example
4619 @code{(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)} will return
4620 @code{x*c^2*z}. This also works with negative powers:
4621 @code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
4622 return @code{x^(-1)*c^2*z}. Note that this only works for multiplications
4623 and not for locating @code{x+y} within @code{x+y+z}.
4626 @node Applying a function on subexpressions, Visitors and tree traversal, Pattern matching and advanced substitutions, Methods and functions
4627 @c node-name, next, previous, up
4628 @section Applying a function on subexpressions
4629 @cindex tree traversal
4630 @cindex @code{map()}
4632 Sometimes you may want to perform an operation on specific parts of an
4633 expression while leaving the general structure of it intact. An example
4634 of this would be a matrix trace operation: the trace of a sum is the sum
4635 of the traces of the individual terms. That is, the trace should @dfn{map}
4636 on the sum, by applying itself to each of the sum's operands. It is possible
4637 to do this manually which usually results in code like this:
4642 if (is_a<matrix>(e))
4643 return ex_to<matrix>(e).trace();
4644 else if (is_a<add>(e)) @{
4646 for (size_t i=0; i<e.nops(); i++)
4647 sum += calc_trace(e.op(i));
4649 @} else if (is_a<mul>)(e)) @{
4657 This is, however, slightly inefficient (if the sum is very large it can take
4658 a long time to add the terms one-by-one), and its applicability is limited to
4659 a rather small class of expressions. If @code{calc_trace()} is called with
4660 a relation or a list as its argument, you will probably want the trace to
4661 be taken on both sides of the relation or of all elements of the list.
4663 GiNaC offers the @code{map()} method to aid in the implementation of such
4667 ex ex::map(map_function & f) const;
4668 ex ex::map(ex (*f)(const ex & e)) const;
4671 In the first (preferred) form, @code{map()} takes a function object that
4672 is subclassed from the @code{map_function} class. In the second form, it
4673 takes a pointer to a function that accepts and returns an expression.
4674 @code{map()} constructs a new expression of the same type, applying the
4675 specified function on all subexpressions (in the sense of @code{op()}),
4678 The use of a function object makes it possible to supply more arguments to
4679 the function that is being mapped, or to keep local state information.
4680 The @code{map_function} class declares a virtual function call operator
4681 that you can overload. Here is a sample implementation of @code{calc_trace()}
4682 that uses @code{map()} in a recursive fashion:
4685 struct calc_trace : public map_function @{
4686 ex operator()(const ex &e)
4688 if (is_a<matrix>(e))
4689 return ex_to<matrix>(e).trace();
4690 else if (is_a<mul>(e)) @{
4693 return e.map(*this);
4698 This function object could then be used like this:
4702 ex M = ... // expression with matrices
4703 calc_trace do_trace;
4704 ex tr = do_trace(M);
4708 Here is another example for you to meditate over. It removes quadratic
4709 terms in a variable from an expanded polynomial:
4712 struct map_rem_quad : public map_function @{
4714 map_rem_quad(const ex & var_) : var(var_) @{@}
4716 ex operator()(const ex & e)
4718 if (is_a<add>(e) || is_a<mul>(e))
4719 return e.map(*this);
4720 else if (is_a<power>(e) &&
4721 e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
4731 symbol x("x"), y("y");
4734 for (int i=0; i<8; i++)
4735 e += pow(x, i) * pow(y, 8-i) * (i+1);
4737 // -> 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
4739 map_rem_quad rem_quad(x);
4740 cout << rem_quad(e) << endl;
4741 // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
4745 @command{ginsh} offers a slightly different implementation of @code{map()}
4746 that allows applying algebraic functions to operands. The second argument
4747 to @code{map()} is an expression containing the wildcard @samp{$0} which
4748 acts as the placeholder for the operands:
4753 > map(a+2*b,sin($0));
4755 > map(@{a,b,c@},$0^2+$0);
4756 @{a^2+a,b^2+b,c^2+c@}
4759 Note that it is only possible to use algebraic functions in the second
4760 argument. You can not use functions like @samp{diff()}, @samp{op()},
4761 @samp{subs()} etc. because these are evaluated immediately:
4764 > map(@{a,b,c@},diff($0,a));
4766 This is because "diff($0,a)" evaluates to "0", so the command is equivalent
4767 to "map(@{a,b,c@},0)".
4771 @node Visitors and tree traversal, Polynomial arithmetic, Applying a function on subexpressions, Methods and functions
4772 @c node-name, next, previous, up
4773 @section Visitors and tree traversal
4774 @cindex tree traversal
4775 @cindex @code{visitor} (class)
4776 @cindex @code{accept()}
4777 @cindex @code{visit()}
4778 @cindex @code{traverse()}
4779 @cindex @code{traverse_preorder()}
4780 @cindex @code{traverse_postorder()}
4782 Suppose that you need a function that returns a list of all indices appearing
4783 in an arbitrary expression. The indices can have any dimension, and for
4784 indices with variance you always want the covariant version returned.
4786 You can't use @code{get_free_indices()} because you also want to include
4787 dummy indices in the list, and you can't use @code{find()} as it needs
4788 specific index dimensions (and it would require two passes: one for indices
4789 with variance, one for plain ones).
4791 The obvious solution to this problem is a tree traversal with a type switch,
4792 such as the following:
4795 void gather_indices_helper(const ex & e, lst & l)
4797 if (is_a<varidx>(e)) @{
4798 const varidx & vi = ex_to<varidx>(e);
4799 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4800 @} else if (is_a<idx>(e)) @{
4803 size_t n = e.nops();
4804 for (size_t i = 0; i < n; ++i)
4805 gather_indices_helper(e.op(i), l);
4809 lst gather_indices(const ex & e)
4812 gather_indices_helper(e, l);
4819 This works fine but fans of object-oriented programming will feel
4820 uncomfortable with the type switch. One reason is that there is a possibility
4821 for subtle bugs regarding derived classes. If we had, for example, written
4824 if (is_a<idx>(e)) @{
4826 @} else if (is_a<varidx>(e)) @{
4830 in @code{gather_indices_helper}, the code wouldn't have worked because the
4831 first line "absorbs" all classes derived from @code{idx}, including
4832 @code{varidx}, so the special case for @code{varidx} would never have been
4835 Also, for a large number of classes, a type switch like the above can get
4836 unwieldy and inefficient (it's a linear search, after all).
4837 @code{gather_indices_helper} only checks for two classes, but if you had to
4838 write a function that required a different implementation for nearly
4839 every GiNaC class, the result would be very hard to maintain and extend.
4841 The cleanest approach to the problem would be to add a new virtual function
4842 to GiNaC's class hierarchy. In our example, there would be specializations
4843 for @code{idx} and @code{varidx} while the default implementation in
4844 @code{basic} performed the tree traversal. Unfortunately, in C++ it's
4845 impossible to add virtual member functions to existing classes without
4846 changing their source and recompiling everything. GiNaC comes with source,
4847 so you could actually do this, but for a small algorithm like the one
4848 presented this would be impractical.
4850 One solution to this dilemma is the @dfn{Visitor} design pattern,
4851 which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
4852 variation, described in detail in
4853 @uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
4854 virtual functions to the class hierarchy to implement operations, GiNaC
4855 provides a single "bouncing" method @code{accept()} that takes an instance
4856 of a special @code{visitor} class and redirects execution to the one
4857 @code{visit()} virtual function of the visitor that matches the type of
4858 object that @code{accept()} was being invoked on.
4860 Visitors in GiNaC must derive from the global @code{visitor} class as well
4861 as from the class @code{T::visitor} of each class @code{T} they want to
4862 visit, and implement the member functions @code{void visit(const T &)} for
4868 void ex::accept(visitor & v) const;
4871 will then dispatch to the correct @code{visit()} member function of the
4872 specified visitor @code{v} for the type of GiNaC object at the root of the
4873 expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
4875 Here is an example of a visitor:
4879 : public visitor, // this is required
4880 public add::visitor, // visit add objects
4881 public numeric::visitor, // visit numeric objects
4882 public basic::visitor // visit basic objects
4884 void visit(const add & x)
4885 @{ cout << "called with an add object" << endl; @}
4887 void visit(const numeric & x)
4888 @{ cout << "called with a numeric object" << endl; @}
4890 void visit(const basic & x)
4891 @{ cout << "called with a basic object" << endl; @}
4895 which can be used as follows:
4906 // prints "called with a numeric object"
4908 // prints "called with an add object"
4910 // prints "called with a basic object"
4914 The @code{visit(const basic &)} method gets called for all objects that are
4915 not @code{numeric} or @code{add} and acts as an (optional) default.
4917 From a conceptual point of view, the @code{visit()} methods of the visitor
4918 behave like a newly added virtual function of the visited hierarchy.
4919 In addition, visitors can store state in member variables, and they can
4920 be extended by deriving a new visitor from an existing one, thus building
4921 hierarchies of visitors.
4923 We can now rewrite our index example from above with a visitor:
4926 class gather_indices_visitor
4927 : public visitor, public idx::visitor, public varidx::visitor
4931 void visit(const idx & i)
4936 void visit(const varidx & vi)
4938 l.append(vi.is_covariant() ? vi : vi.toggle_variance());
4942 const lst & get_result() // utility function
4951 What's missing is the tree traversal. We could implement it in
4952 @code{visit(const basic &)}, but GiNaC has predefined methods for this:
4955 void ex::traverse_preorder(visitor & v) const;
4956 void ex::traverse_postorder(visitor & v) const;
4957 void ex::traverse(visitor & v) const;
4960 @code{traverse_preorder()} visits a node @emph{before} visiting its
4961 subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
4962 visiting its subexpressions. @code{traverse()} is a synonym for
4963 @code{traverse_preorder()}.
4965 Here is a new implementation of @code{gather_indices()} that uses the visitor
4966 and @code{traverse()}:
4969 lst gather_indices(const ex & e)
4971 gather_indices_visitor v;
4973 return v.get_result();
4977 Alternatively, you could use pre- or postorder iterators for the tree
4981 lst gather_indices(const ex & e)
4983 gather_indices_visitor v;
4984 for (const_preorder_iterator i = e.preorder_begin();
4985 i != e.preorder_end(); ++i) @{
4988 return v.get_result();
4993 @node Polynomial arithmetic, Rational expressions, Visitors and tree traversal, Methods and functions
4994 @c node-name, next, previous, up
4995 @section Polynomial arithmetic
4997 @subsection Testing whether an expression is a polynomial
4998 @cindex @code{is_polynomial()}
5000 Testing whether an expression is a polynomial in one or more variables
5001 can be done with the method
5003 bool ex::is_polynomial(const ex & vars) const;
5005 In the case of more than
5006 one variable, the variables are given as a list.
5009 (x*y*sin(y)).is_polynomial(x) // Returns true.
5010 (x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
5013 @subsection Expanding and collecting
5014 @cindex @code{expand()}
5015 @cindex @code{collect()}
5016 @cindex @code{collect_common_factors()}
5018 A polynomial in one or more variables has many equivalent
5019 representations. Some useful ones serve a specific purpose. Consider
5020 for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
5021 21*y*z + 4*z^2} (written down here in output-style). It is equivalent
5022 to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
5023 representations are the recursive ones where one collects for exponents
5024 in one of the three variable. Since the factors are themselves
5025 polynomials in the remaining two variables the procedure can be
5026 repeated. In our example, two possibilities would be @math{(4*y + z)*x
5027 + 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
5030 To bring an expression into expanded form, its method
5033 ex ex::expand(unsigned options = 0);
5036 may be called. In our example above, this corresponds to @math{4*x*y +
5037 x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
5038 GiNaC is not easy to guess you should be prepared to see different
5039 orderings of terms in such sums!
5041 Another useful representation of multivariate polynomials is as a
5042 univariate polynomial in one of the variables with the coefficients
5043 being polynomials in the remaining variables. The method
5044 @code{collect()} accomplishes this task:
5047 ex ex::collect(const ex & s, bool distributed = false);
5050 The first argument to @code{collect()} can also be a list of objects in which
5051 case the result is either a recursively collected polynomial, or a polynomial
5052 in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
5053 by the @code{distributed} flag.
5055 Note that the original polynomial needs to be in expanded form (for the
5056 variables concerned) in order for @code{collect()} to be able to find the
5057 coefficients properly.
5059 The following @command{ginsh} transcript shows an application of @code{collect()}
5060 together with @code{find()}:
5063 > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
5064 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)
5065 +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
5066 > collect(a,@{p,q@});
5067 d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
5068 +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
5069 > collect(a,find(a,sin($1)));
5070 (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
5071 > collect(a,@{find(a,sin($1)),p,q@});
5072 (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
5073 > collect(a,@{find(a,sin($1)),d@});
5074 (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
5077 Polynomials can often be brought into a more compact form by collecting
5078 common factors from the terms of sums. This is accomplished by the function
5081 ex collect_common_factors(const ex & e);
5084 This function doesn't perform a full factorization but only looks for
5085 factors which are already explicitly present:
5088 > collect_common_factors(a*x+a*y);
5090 > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
5092 > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
5093 (c+a)*a*(x*y+y^2+x)*b
5096 @subsection Degree and coefficients
5097 @cindex @code{degree()}
5098 @cindex @code{ldegree()}
5099 @cindex @code{coeff()}
5101 The degree and low degree of a polynomial can be obtained using the two
5105 int ex::degree(const ex & s);
5106 int ex::ldegree(const ex & s);
5109 which also work reliably on non-expanded input polynomials (they even work
5110 on rational functions, returning the asymptotic degree). By definition, the
5111 degree of zero is zero. To extract a coefficient with a certain power from
5112 an expanded polynomial you use
5115 ex ex::coeff(const ex & s, int n);
5118 You can also obtain the leading and trailing coefficients with the methods
5121 ex ex::lcoeff(const ex & s);
5122 ex ex::tcoeff(const ex & s);
5125 which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
5128 An application is illustrated in the next example, where a multivariate
5129 polynomial is analyzed:
5133 symbol x("x"), y("y");
5134 ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
5135 - pow(x+y,2) + 2*pow(y+2,2) - 8;
5136 ex Poly = PolyInp.expand();
5138 for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
5139 cout << "The x^" << i << "-coefficient is "
5140 << Poly.coeff(x,i) << endl;
5142 cout << "As polynomial in y: "
5143 << Poly.collect(y) << endl;
5147 When run, it returns an output in the following fashion:
5150 The x^0-coefficient is y^2+11*y
5151 The x^1-coefficient is 5*y^2-2*y
5152 The x^2-coefficient is -1
5153 The x^3-coefficient is 4*y
5154 As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
5157 As always, the exact output may vary between different versions of GiNaC
5158 or even from run to run since the internal canonical ordering is not
5159 within the user's sphere of influence.
5161 @code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
5162 @code{tcoeff()} and @code{collect()} can also be used to a certain degree
5163 with non-polynomial expressions as they not only work with symbols but with
5164 constants, functions and indexed objects as well:
5168 symbol a("a"), b("b"), c("c"), x("x");
5169 idx i(symbol("i"), 3);
5171 ex e = pow(sin(x) - cos(x), 4);
5172 cout << e.degree(cos(x)) << endl;
5174 cout << e.expand().coeff(sin(x), 3) << endl;
5177 e = indexed(a+b, i) * indexed(b+c, i);
5178 e = e.expand(expand_options::expand_indexed);
5179 cout << e.collect(indexed(b, i)) << endl;
5180 // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
5185 @subsection Polynomial division
5186 @cindex polynomial division
5189 @cindex pseudo-remainder
5190 @cindex @code{quo()}
5191 @cindex @code{rem()}
5192 @cindex @code{prem()}
5193 @cindex @code{divide()}
5198 ex quo(const ex & a, const ex & b, const ex & x);
5199 ex rem(const ex & a, const ex & b, const ex & x);
5202 compute the quotient and remainder of univariate polynomials in the variable
5203 @samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
5205 The additional function
5208 ex prem(const ex & a, const ex & b, const ex & x);
5211 computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
5212 @math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
5214 Exact division of multivariate polynomials is performed by the function
5217 bool divide(const ex & a, const ex & b, ex & q);
5220 If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
5221 and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
5222 in which case the value of @code{q} is undefined.
5225 @subsection Unit, content and primitive part
5226 @cindex @code{unit()}
5227 @cindex @code{content()}
5228 @cindex @code{primpart()}
5229 @cindex @code{unitcontprim()}
5234 ex ex::unit(const ex & x);
5235 ex ex::content(const ex & x);
5236 ex ex::primpart(const ex & x);
5237 ex ex::primpart(const ex & x, const ex & c);
5240 return the unit part, content part, and primitive polynomial of a multivariate
5241 polynomial with respect to the variable @samp{x} (the unit part being the sign
5242 of the leading coefficient, the content part being the GCD of the coefficients,
5243 and the primitive polynomial being the input polynomial divided by the unit and
5244 content parts). The second variant of @code{primpart()} expects the previously
5245 calculated content part of the polynomial in @code{c}, which enables it to
5246 work faster in the case where the content part has already been computed. The
5247 product of unit, content, and primitive part is the original polynomial.
5249 Additionally, the method
5252 void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
5255 computes the unit, content, and primitive parts in one go, returning them
5256 in @code{u}, @code{c}, and @code{p}, respectively.
5259 @subsection GCD, LCM and resultant
5262 @cindex @code{gcd()}
5263 @cindex @code{lcm()}
5265 The functions for polynomial greatest common divisor and least common
5266 multiple have the synopsis
5269 ex gcd(const ex & a, const ex & b);
5270 ex lcm(const ex & a, const ex & b);
5273 The functions @code{gcd()} and @code{lcm()} accept two expressions
5274 @code{a} and @code{b} as arguments and return a new expression, their
5275 greatest common divisor or least common multiple, respectively. If the
5276 polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
5277 and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
5278 the coefficients must be rationals.
5281 #include <ginac/ginac.h>
5282 using namespace GiNaC;
5286 symbol x("x"), y("y"), z("z");
5287 ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
5288 ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
5290 ex P_gcd = gcd(P_a, P_b);
5292 ex P_lcm = lcm(P_a, P_b);
5293 // 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
5298 @cindex @code{resultant()}
5300 The resultant of two expressions only makes sense with polynomials.
5301 It is always computed with respect to a specific symbol within the
5302 expressions. The function has the interface
5305 ex resultant(const ex & a, const ex & b, const ex & s);
5308 Resultants are symmetric in @code{a} and @code{b}. The following example
5309 computes the resultant of two expressions with respect to @code{x} and
5310 @code{y}, respectively:
5313 #include <ginac/ginac.h>
5314 using namespace GiNaC;
5318 symbol x("x"), y("y");
5320 ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
5323 r = resultant(e1, e2, x);
5325 r = resultant(e1, e2, y);
5330 @subsection Square-free decomposition
5331 @cindex square-free decomposition
5332 @cindex factorization
5333 @cindex @code{sqrfree()}
5335 GiNaC still lacks proper factorization support. Some form of
5336 factorization is, however, easily implemented by noting that factors
5337 appearing in a polynomial with power two or more also appear in the
5338 derivative and hence can easily be found by computing the GCD of the
5339 original polynomial and its derivatives. Any decent system has an
5340 interface for this so called square-free factorization. So we provide
5343 ex sqrfree(const ex & a, const lst & l = lst());
5345 Here is an example that by the way illustrates how the exact form of the
5346 result may slightly depend on the order of differentiation, calling for
5347 some care with subsequent processing of the result:
5350 symbol x("x"), y("y");
5351 ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
5353 cout << sqrfree(BiVarPol, lst(x,y)) << endl;
5354 // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
5356 cout << sqrfree(BiVarPol, lst(y,x)) << endl;
5357 // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
5359 cout << sqrfree(BiVarPol) << endl;
5360 // -> depending on luck, any of the above
5363 Note also, how factors with the same exponents are not fully factorized
5367 @node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
5368 @c node-name, next, previous, up
5369 @section Rational expressions
5371 @subsection The @code{normal} method
5372 @cindex @code{normal()}
5373 @cindex simplification
5374 @cindex temporary replacement
5376 Some basic form of simplification of expressions is called for frequently.
5377 GiNaC provides the method @code{.normal()}, which converts a rational function
5378 into an equivalent rational function of the form @samp{numerator/denominator}
5379 where numerator and denominator are coprime. If the input expression is already
5380 a fraction, it just finds the GCD of numerator and denominator and cancels it,
5381 otherwise it performs fraction addition and multiplication.
5383 @code{.normal()} can also be used on expressions which are not rational functions
5384 as it will replace all non-rational objects (like functions or non-integer
5385 powers) by temporary symbols to bring the expression to the domain of rational
5386 functions before performing the normalization, and re-substituting these
5387 symbols afterwards. This algorithm is also available as a separate method
5388 @code{.to_rational()}, described below.
5390 This means that both expressions @code{t1} and @code{t2} are indeed
5391 simplified in this little code snippet:
5396 ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
5397 ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
5398 std::cout << "t1 is " << t1.normal() << std::endl;
5399 std::cout << "t2 is " << t2.normal() << std::endl;
5403 Of course this works for multivariate polynomials too, so the ratio of
5404 the sample-polynomials from the section about GCD and LCM above would be
5405 normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
5408 @subsection Numerator and denominator
5411 @cindex @code{numer()}
5412 @cindex @code{denom()}
5413 @cindex @code{numer_denom()}
5415 The numerator and denominator of an expression can be obtained with
5420 ex ex::numer_denom();
5423 These functions will first normalize the expression as described above and
5424 then return the numerator, denominator, or both as a list, respectively.
5425 If you need both numerator and denominator, calling @code{numer_denom()} is
5426 faster than using @code{numer()} and @code{denom()} separately.
5429 @subsection Converting to a polynomial or rational expression
5430 @cindex @code{to_polynomial()}
5431 @cindex @code{to_rational()}
5433 Some of the methods described so far only work on polynomials or rational
5434 functions. GiNaC provides a way to extend the domain of these functions to
5435 general expressions by using the temporary replacement algorithm described
5436 above. You do this by calling
5439 ex ex::to_polynomial(exmap & m);
5440 ex ex::to_polynomial(lst & l);
5444 ex ex::to_rational(exmap & m);
5445 ex ex::to_rational(lst & l);
5448 on the expression to be converted. The supplied @code{exmap} or @code{lst}
5449 will be filled with the generated temporary symbols and their replacement
5450 expressions in a format that can be used directly for the @code{subs()}
5451 method. It can also already contain a list of replacements from an earlier
5452 application of @code{.to_polynomial()} or @code{.to_rational()}, so it's
5453 possible to use it on multiple expressions and get consistent results.
5455 The difference between @code{.to_polynomial()} and @code{.to_rational()}
5456 is probably best illustrated with an example:
5460 symbol x("x"), y("y");
5461 ex a = 2*x/sin(x) - y/(3*sin(x));
5465 ex p = a.to_polynomial(lp);
5466 cout << " = " << p << "\n with " << lp << endl;
5467 // = symbol3*symbol2*y+2*symbol2*x
5468 // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
5471 ex r = a.to_rational(lr);
5472 cout << " = " << r << "\n with " << lr << endl;
5473 // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
5474 // with @{symbol4==sin(x)@}
5478 The following more useful example will print @samp{sin(x)-cos(x)}:
5483 ex a = pow(sin(x), 2) - pow(cos(x), 2);
5484 ex b = sin(x) + cos(x);
5487 divide(a.to_polynomial(m), b.to_polynomial(m), q);
5488 cout << q.subs(m) << endl;
5493 @node Symbolic differentiation, Series expansion, Rational expressions, Methods and functions
5494 @c node-name, next, previous, up
5495 @section Symbolic differentiation
5496 @cindex differentiation
5497 @cindex @code{diff()}
5499 @cindex product rule
5501 GiNaC's objects know how to differentiate themselves. Thus, a
5502 polynomial (class @code{add}) knows that its derivative is the sum of
5503 the derivatives of all the monomials:
5507 symbol x("x"), y("y"), z("z");
5508 ex P = pow(x, 5) + pow(x, 2) + y;
5510 cout << P.diff(x,2) << endl;
5512 cout << P.diff(y) << endl; // 1
5514 cout << P.diff(z) << endl; // 0
5519 If a second integer parameter @var{n} is given, the @code{diff} method
5520 returns the @var{n}th derivative.
5522 If @emph{every} object and every function is told what its derivative
5523 is, all derivatives of composed objects can be calculated using the
5524 chain rule and the product rule. Consider, for instance the expression
5525 @code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
5526 @code{sinh(x)} and the derivative of @code{pow(x,-1)} is
5527 @code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
5528 out that the composition is the generating function for Euler Numbers,
5529 i.e. the so called @var{n}th Euler number is the coefficient of
5530 @code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
5531 identity to code a function that generates Euler numbers in just three
5534 @cindex Euler numbers
5536 #include <ginac/ginac.h>
5537 using namespace GiNaC;
5539 ex EulerNumber(unsigned n)
5542 const ex generator = pow(cosh(x),-1);
5543 return generator.diff(x,n).subs(x==0);
5548 for (unsigned i=0; i<11; i+=2)
5549 std::cout << EulerNumber(i) << std::endl;
5554 When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
5555 @code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
5556 @code{i} by two since all odd Euler numbers vanish anyways.
5559 @node Series expansion, Symmetrization, Symbolic differentiation, Methods and functions
5560 @c node-name, next, previous, up
5561 @section Series expansion
5562 @cindex @code{series()}
5563 @cindex Taylor expansion
5564 @cindex Laurent expansion
5565 @cindex @code{pseries} (class)
5566 @cindex @code{Order()}
5568 Expressions know how to expand themselves as a Taylor series or (more
5569 generally) a Laurent series. As in most conventional Computer Algebra
5570 Systems, no distinction is made between those two. There is a class of
5571 its own for storing such series (@code{class pseries}) and a built-in
5572 function (called @code{Order}) for storing the order term of the series.
5573 As a consequence, if you want to work with series, i.e. multiply two
5574 series, you need to call the method @code{ex::series} again to convert
5575 it to a series object with the usual structure (expansion plus order
5576 term). A sample application from special relativity could read:
5579 #include <ginac/ginac.h>
5580 using namespace std;
5581 using namespace GiNaC;
5585 symbol v("v"), c("c");
5587 ex gamma = 1/sqrt(1 - pow(v/c,2));
5588 ex mass_nonrel = gamma.series(v==0, 10);
5590 cout << "the relativistic mass increase with v is " << endl
5591 << mass_nonrel << endl;
5593 cout << "the inverse square of this series is " << endl
5594 << pow(mass_nonrel,-2).series(v==0, 10) << endl;
5598 Only calling the series method makes the last output simplify to
5599 @math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
5600 series raised to the power @math{-2}.
5602 @cindex Machin's formula
5603 As another instructive application, let us calculate the numerical
5604 value of Archimedes' constant
5608 (for which there already exists the built-in constant @code{Pi})
5609 using John Machin's amazing formula
5611 $\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
5614 @math{Pi==16*atan(1/5)-4*atan(1/239)}.
5616 This equation (and similar ones) were used for over 200 years for
5617 computing digits of pi (see @cite{Pi Unleashed}). We may expand the
5618 arcus tangent around @code{0} and insert the fractions @code{1/5} and
5619 @code{1/239}. However, as we have seen, a series in GiNaC carries an
5620 order term with it and the question arises what the system is supposed
5621 to do when the fractions are plugged into that order term. The solution
5622 is to use the function @code{series_to_poly()} to simply strip the order
5626 #include <ginac/ginac.h>
5627 using namespace GiNaC;
5629 ex machin_pi(int degr)
5632 ex pi_expansion = series_to_poly(atan(x).series(x,degr));
5633 ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
5634 -4*pi_expansion.subs(x==numeric(1,239));
5640 using std::cout; // just for fun, another way of...
5641 using std::endl; // ...dealing with this namespace std.
5643 for (int i=2; i<12; i+=2) @{
5644 pi_frac = machin_pi(i);
5645 cout << i << ":\t" << pi_frac << endl
5646 << "\t" << pi_frac.evalf() << endl;
5652 Note how we just called @code{.series(x,degr)} instead of
5653 @code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
5654 method @code{series()}: if the first argument is a symbol the expression
5655 is expanded in that symbol around point @code{0}. When you run this
5656 program, it will type out:
5660 3.1832635983263598326
5661 4: 5359397032/1706489875
5662 3.1405970293260603143
5663 6: 38279241713339684/12184551018734375
5664 3.141621029325034425
5665 8: 76528487109180192540976/24359780855939418203125
5666 3.141591772182177295
5667 10: 327853873402258685803048818236/104359128170408663038552734375
5668 3.1415926824043995174
5672 @node Symmetrization, Built-in functions, Series expansion, Methods and functions
5673 @c node-name, next, previous, up
5674 @section Symmetrization
5675 @cindex @code{symmetrize()}
5676 @cindex @code{antisymmetrize()}
5677 @cindex @code{symmetrize_cyclic()}
5682 ex ex::symmetrize(const lst & l);
5683 ex ex::antisymmetrize(const lst & l);
5684 ex ex::symmetrize_cyclic(const lst & l);
5687 symmetrize an expression by returning the sum over all symmetric,
5688 antisymmetric or cyclic permutations of the specified list of objects,
5689 weighted by the number of permutations.
5691 The three additional methods
5694 ex ex::symmetrize();
5695 ex ex::antisymmetrize();
5696 ex ex::symmetrize_cyclic();
5699 symmetrize or antisymmetrize an expression over its free indices.
5701 Symmetrization is most useful with indexed expressions but can be used with
5702 almost any kind of object (anything that is @code{subs()}able):
5706 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
5707 symbol A("A"), B("B"), a("a"), b("b"), c("c");
5709 cout << indexed(A, i, j).symmetrize() << endl;
5710 // -> 1/2*A.j.i+1/2*A.i.j
5711 cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
5712 // -> -1/2*A.j.i.k+1/2*A.i.j.k
5713 cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
5714 // -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
5718 @node Built-in functions, Multiple polylogarithms, Symmetrization, Methods and functions
5719 @c node-name, next, previous, up
5720 @section Predefined mathematical functions
5722 @subsection Overview
5724 GiNaC contains the following predefined mathematical functions:
5727 @multitable @columnfractions .30 .70
5728 @item @strong{Name} @tab @strong{Function}
5731 @cindex @code{abs()}
5732 @item @code{step(x)}
5734 @cindex @code{step()}
5735 @item @code{csgn(x)}
5737 @cindex @code{conjugate()}
5738 @item @code{conjugate(x)}
5739 @tab complex conjugation
5740 @cindex @code{real_part()}
5741 @item @code{real_part(x)}
5743 @cindex @code{imag_part()}
5744 @item @code{imag_part(x)}
5746 @item @code{sqrt(x)}
5747 @tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
5748 @cindex @code{sqrt()}
5751 @cindex @code{sin()}
5754 @cindex @code{cos()}
5757 @cindex @code{tan()}
5758 @item @code{asin(x)}
5760 @cindex @code{asin()}
5761 @item @code{acos(x)}
5763 @cindex @code{acos()}
5764 @item @code{atan(x)}
5765 @tab inverse tangent
5766 @cindex @code{atan()}
5767 @item @code{atan2(y, x)}
5768 @tab inverse tangent with two arguments
5769 @item @code{sinh(x)}
5770 @tab hyperbolic sine
5771 @cindex @code{sinh()}
5772 @item @code{cosh(x)}
5773 @tab hyperbolic cosine
5774 @cindex @code{cosh()}
5775 @item @code{tanh(x)}
5776 @tab hyperbolic tangent
5777 @cindex @code{tanh()}
5778 @item @code{asinh(x)}
5779 @tab inverse hyperbolic sine
5780 @cindex @code{asinh()}
5781 @item @code{acosh(x)}
5782 @tab inverse hyperbolic cosine
5783 @cindex @code{acosh()}
5784 @item @code{atanh(x)}
5785 @tab inverse hyperbolic tangent
5786 @cindex @code{atanh()}
5788 @tab exponential function
5789 @cindex @code{exp()}
5791 @tab natural logarithm
5792 @cindex @code{log()}
5795 @cindex @code{Li2()}
5796 @item @code{Li(m, x)}
5797 @tab classical polylogarithm as well as multiple polylogarithm
5799 @item @code{G(a, y)}
5800 @tab multiple polylogarithm
5802 @item @code{G(a, s, y)}
5803 @tab multiple polylogarithm with explicit signs for the imaginary parts
5805 @item @code{S(n, p, x)}
5806 @tab Nielsen's generalized polylogarithm
5808 @item @code{H(m, x)}
5809 @tab harmonic polylogarithm
5811 @item @code{zeta(m)}
5812 @tab Riemann's zeta function as well as multiple zeta value
5813 @cindex @code{zeta()}
5814 @item @code{zeta(m, s)}
5815 @tab alternating Euler sum
5816 @cindex @code{zeta()}
5817 @item @code{zetaderiv(n, x)}
5818 @tab derivatives of Riemann's zeta function
5819 @item @code{tgamma(x)}
5821 @cindex @code{tgamma()}
5822 @cindex gamma function
5823 @item @code{lgamma(x)}
5824 @tab logarithm of gamma function
5825 @cindex @code{lgamma()}
5826 @item @code{beta(x, y)}
5827 @tab beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
5828 @cindex @code{beta()}
5830 @tab psi (digamma) function
5831 @cindex @code{psi()}
5832 @item @code{psi(n, x)}
5833 @tab derivatives of psi function (polygamma functions)
5834 @item @code{factorial(n)}
5835 @tab factorial function @math{n!}
5836 @cindex @code{factorial()}
5837 @item @code{binomial(n, k)}
5838 @tab binomial coefficients
5839 @cindex @code{binomial()}
5840 @item @code{Order(x)}
5841 @tab order term function in truncated power series
5842 @cindex @code{Order()}
5847 For functions that have a branch cut in the complex plane GiNaC follows
5848 the conventions for C++ as defined in the ANSI standard as far as
5849 possible. In particular: the natural logarithm (@code{log}) and the
5850 square root (@code{sqrt}) both have their branch cuts running along the
5851 negative real axis where the points on the axis itself belong to the
5852 upper part (i.e. continuous with quadrant II). The inverse
5853 trigonometric and hyperbolic functions are not defined for complex
5854 arguments by the C++ standard, however. In GiNaC we follow the
5855 conventions used by CLN, which in turn follow the carefully designed
5856 definitions in the Common Lisp standard. It should be noted that this
5857 convention is identical to the one used by the C99 standard and by most
5858 serious CAS. It is to be expected that future revisions of the C++
5859 standard incorporate these functions in the complex domain in a manner
5860 compatible with C99.
5862 @node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
5863 @c node-name, next, previous, up
5864 @subsection Multiple polylogarithms
5866 @cindex polylogarithm
5867 @cindex Nielsen's generalized polylogarithm
5868 @cindex harmonic polylogarithm
5869 @cindex multiple zeta value
5870 @cindex alternating Euler sum
5871 @cindex multiple polylogarithm
5873 The multiple polylogarithm is the most generic member of a family of functions,
5874 to which others like the harmonic polylogarithm, Nielsen's generalized
5875 polylogarithm and the multiple zeta value belong.
5876 Everyone of these functions can also be written as a multiple polylogarithm with specific
5877 parameters. This whole family of functions is therefore often referred to simply as
5878 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
5879 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
5880 @code{Li} and @code{G} in principle represent the same function, the different
5881 notations are more natural to the series representation or the integral
5882 representation, respectively.
5884 To facilitate the discussion of these functions we distinguish between indices and
5885 arguments as parameters. In the table above indices are printed as @code{m}, @code{s},
5886 @code{n} or @code{p}, whereas arguments are printed as @code{x}, @code{a} and @code{y}.
5888 To define a @code{Li}, @code{H} or @code{zeta} with a depth greater than one, you have to
5889 pass a GiNaC @code{lst} for the indices @code{m} and @code{s}, and in the case of @code{Li}
5890 for the argument @code{x} as well. The parameter @code{a} of @code{G} must always be a @code{lst} containing
5891 the arguments in expanded form. If @code{G} is used with a third parameter @code{s}, @code{s} must
5892 have the same length as @code{a}. It contains then the signs of the imaginary parts of the arguments. If
5893 @code{s} is not given, the signs default to +1.
5894 Note that @code{Li} and @code{zeta} are polymorphic in this respect. They can stand in for
5895 the classical polylogarithm and Riemann's zeta function (if depth is one), as well as for
5896 the multiple polylogarithm and the multiple zeta value, respectively. Note also, that
5897 GiNaC doesn't check whether the @code{lst}s for two parameters do have the same length.
5898 It is up to the user to ensure this, otherwise evaluating will result in undefined behavior.
5900 The functions print in LaTeX format as
5902 ${\rm Li\;\!}_{m_1,m_2,\ldots,m_k}(x_1,x_2,\ldots,x_k)$,
5908 ${\rm H\;\!}_{m_1,m_2,\ldots,m_k}(x)$ and
5911 $\zeta(m_1,m_2,\ldots,m_k)$.
5913 If @code{zeta} is an alternating zeta sum, i.e. @code{zeta(m,s)}, the indices with negative sign
5914 are printed with a line above, e.g.
5916 $\zeta(5,\overline{2})$.
5918 The order of indices and arguments in the GiNaC @code{lst}s and in the output is the same.
5920 Definitions and analytical as well as numerical properties of multiple polylogarithms
5921 are too numerous to be covered here. Instead, the user is referred to the publications listed at the
5922 end of this section. The implementation in GiNaC adheres to the definitions and conventions therein,
5923 except for a few differences which will be explicitly stated in the following.
5925 One difference is about the order of the indices and arguments. For GiNaC we adopt the convention
5926 that the indices and arguments are understood to be in the same order as in which they appear in
5927 the series representation. This means
5929 ${\rm Li\;\!}_{m_1,m_2,m_3}(x,1,1) = {\rm H\;\!}_{m_1,m_2,m_3}(x)$ and
5932 ${\rm Li\;\!}_{2,1}(1,1) = \zeta(2,1) = \zeta(3)$, but
5935 $\zeta(1,2)$ evaluates to infinity.
5937 So in comparison to the referenced publications the order of indices and arguments for @code{Li}
5940 The functions only evaluate if the indices are integers greater than zero, except for the indices
5941 @code{s} in @code{zeta} and @code{G} as well as @code{m} in @code{H}. Since @code{s}
5942 will be interpreted as the sequence of signs for the corresponding indices
5943 @code{m} or the sign of the imaginary part for the
5944 corresponding arguments @code{a}, it must contain 1 or -1, e.g.
5945 @code{zeta(lst(3,4), lst(-1,1))} means
5947 $\zeta(\overline{3},4)$
5950 @code{G(lst(a,b), lst(-1,1), c)} means
5952 $G(a-0\epsilon,b+0\epsilon;c)$.
5954 The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
5955 be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
5956 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
5957 indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
5958 the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
5959 Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
5960 evaluates also for negative integers and positive even integers. For example:
5963 > Li(@{3,1@},@{x,1@});
5966 -zeta(@{3,2@},@{-1,-1@})
5971 It is easy to tell for a given function into which other function it can be rewritten, may
5972 it be a less-generic or a more-generic one, except for harmonic polylogarithms @code{H}
5973 with negative indices or trailing zeros (the example above gives a hint). Signs can
5974 quickly be messed up, for example. Therefore GiNaC offers a C++ function
5975 @code{convert_H_to_Li()} to deal with the upgrade of a @code{H} to a multiple polylogarithm
5976 @code{Li} (@code{eval()} already cares for the possible downgrade):
5979 > convert_H_to_Li(@{0,-2,-1,3@},x);
5980 Li(@{3,1,3@},@{-x,1,-1@})
5981 > convert_H_to_Li(@{2,-1,0@},x);
5982 -Li(@{2,1@},@{x,-1@})*log(x)+2*Li(@{3,1@},@{x,-1@})+Li(@{2,2@},@{x,-1@})
5985 Every function can be numerically evaluated for
5986 arbitrary real or complex arguments. The precision is arbitrary and can be set through the
5987 global variable @code{Digits}:
5992 > evalf(zeta(@{3,1,3,1@}));
5993 0.005229569563530960100930652283899231589890420784634635522547448972148869544...
5996 Note that the convention for arguments on the branch cut in GiNaC as stated above is
5997 different from the one Remiddi and Vermaseren have chosen for the harmonic polylogarithm.
5999 If a function evaluates to infinity, no exceptions are raised, but the function is returned
6004 In long expressions this helps a lot with debugging, because you can easily spot
6005 the divergencies. But on the other hand, you have to make sure for yourself, that no illegal
6006 cancellations of divergencies happen.
6008 Useful publications:
6010 @cite{Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals},
6011 S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083
6013 @cite{Harmonic Polylogarithms},
6014 E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
6016 @cite{Special Values of Multiple Polylogarithms},
6017 J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
6019 @cite{Numerical Evaluation of Multiple Polylogarithms},
6020 J.Vollinga, S.Weinzierl, hep-ph/0410259
6022 @node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
6023 @c node-name, next, previous, up
6024 @section Complex expressions
6026 @cindex @code{conjugate()}
6028 For dealing with complex expressions there are the methods
6036 that return respectively the complex conjugate, the real part and the
6037 imaginary part of an expression. Complex conjugation works as expected
6038 for all built-in functinos and objects. Taking real and imaginary
6039 parts has not yet been implemented for all built-in functions. In cases where
6040 it is not known how to conjugate or take a real/imaginary part one
6041 of the functions @code{conjugate}, @code{real_part} or @code{imag_part}
6042 is returned. For instance, in case of a complex symbol @code{x}
6043 (symbols are complex by default), one could not simplify
6044 @code{conjugate(x)}. In the case of strings of gamma matrices,
6045 the @code{conjugate} method takes the Dirac conjugate.
6050 varidx a(symbol("a"), 4), b(symbol("b"), 4);
6054 cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
6055 // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
6056 cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
6057 // -> -gamma5*gamma~b*gamma~a
6061 If you declare your own GiNaC functions, then they will conjugate themselves
6062 by conjugating their arguments. This is the default strategy. If you want to
6063 change this behavior, you have to supply a specialized conjugation method
6064 for your function (see @ref{Symbolic functions} and the GiNaC source-code
6065 for @code{abs} as an example). Also, specialized methods can be provided
6066 to take real and imaginary parts of user-defined functions.
6068 @node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
6069 @c node-name, next, previous, up
6070 @section Solving linear systems of equations
6071 @cindex @code{lsolve()}
6073 The function @code{lsolve()} provides a convenient wrapper around some
6074 matrix operations that comes in handy when a system of linear equations
6078 ex lsolve(const ex & eqns, const ex & symbols,
6079 unsigned options = solve_algo::automatic);
6082 Here, @code{eqns} is a @code{lst} of equalities (i.e. class
6083 @code{relational}) while @code{symbols} is a @code{lst} of
6084 indeterminates. (@xref{The class hierarchy}, for an exposition of class
6087 It returns the @code{lst} of solutions as an expression. As an example,
6088 let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
6092 symbol a("a"), b("b"), x("x"), y("y");
6094 eqns = a*x+b*y==3, x-y==b;
6096 cout << lsolve(eqns, vars) << endl;
6097 // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
6100 When the linear equations @code{eqns} are underdetermined, the solution
6101 will contain one or more tautological entries like @code{x==x},
6102 depending on the rank of the system. When they are overdetermined, the
6103 solution will be an empty @code{lst}. Note the third optional parameter
6104 to @code{lsolve()}: it accepts the same parameters as
6105 @code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
6109 @node Input/output, Extending GiNaC, Solving linear systems of equations, Methods and functions
6110 @c node-name, next, previous, up
6111 @section Input and output of expressions
6114 @subsection Expression output
6116 @cindex output of expressions
6118 Expressions can simply be written to any stream:
6123 ex e = 4.5*I+pow(x,2)*3/2;
6124 cout << e << endl; // prints '4.5*I+3/2*x^2'
6128 The default output format is identical to the @command{ginsh} input syntax and
6129 to that used by most computer algebra systems, but not directly pastable
6130 into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
6131 is printed as @samp{x^2}).
6133 It is possible to print expressions in a number of different formats with
6134 a set of stream manipulators;
6137 std::ostream & dflt(std::ostream & os);
6138 std::ostream & latex(std::ostream & os);
6139 std::ostream & tree(std::ostream & os);
6140 std::ostream & csrc(std::ostream & os);
6141 std::ostream & csrc_float(std::ostream & os);
6142 std::ostream & csrc_double(std::ostream & os);
6143 std::ostream & csrc_cl_N(std::ostream & os);
6144 std::ostream & index_dimensions(std::ostream & os);
6145 std::ostream & no_index_dimensions(std::ostream & os);
6148 The @code{tree}, @code{latex} and @code{csrc} formats are also available in
6149 @command{ginsh} via the @code{print()}, @code{print_latex()} and
6150 @code{print_csrc()} functions, respectively.
6153 All manipulators affect the stream state permanently. To reset the output
6154 format to the default, use the @code{dflt} manipulator:
6158 cout << latex; // all output to cout will be in LaTeX format from
6160 cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6161 cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
6162 cout << dflt; // revert to default output format
6163 cout << e << endl; // prints '4.5*I+3/2*x^2'
6167 If you don't want to affect the format of the stream you're working with,
6168 you can output to a temporary @code{ostringstream} like this:
6173 s << latex << e; // format of cout remains unchanged
6174 cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
6179 @cindex @code{csrc_float}
6180 @cindex @code{csrc_double}
6181 @cindex @code{csrc_cl_N}
6182 The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
6183 @code{csrc_double} and @code{csrc_cl_N} manipulators set the output to a
6184 format that can be directly used in a C or C++ program. The three possible
6185 formats select the data types used for numbers (@code{csrc_cl_N} uses the
6186 classes provided by the CLN library):
6190 cout << "f = " << csrc_float << e << ";\n";
6191 cout << "d = " << csrc_double << e << ";\n";
6192 cout << "n = " << csrc_cl_N << e << ";\n";
6196 The above example will produce (note the @code{x^2} being converted to
6200 f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
6201 d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
6202 n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));
6206 The @code{tree} manipulator allows dumping the internal structure of an
6207 expression for debugging purposes:
6218 add, hash=0x0, flags=0x3, nops=2
6219 power, hash=0x0, flags=0x3, nops=2
6220 x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
6221 2 (numeric), hash=0x6526b0fa, flags=0xf
6222 3/2 (numeric), hash=0xf9828fbd, flags=0xf
6225 4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
6229 @cindex @code{latex}
6230 The @code{latex} output format is for LaTeX parsing in mathematical mode.
6231 It is rather similar to the default format but provides some braces needed
6232 by LaTeX for delimiting boxes and also converts some common objects to
6233 conventional LaTeX names. It is possible to give symbols a special name for
6234 LaTeX output by supplying it as a second argument to the @code{symbol}
6237 For example, the code snippet
6241 symbol x("x", "\\circ");
6242 ex e = lgamma(x).series(x==0,3);
6243 cout << latex << e << endl;
6250 @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
6251 +\mathcal@{O@}(\circ^@{3@})
6254 @cindex @code{index_dimensions}
6255 @cindex @code{no_index_dimensions}
6256 Index dimensions are normally hidden in the output. To make them visible, use
6257 the @code{index_dimensions} manipulator. The dimensions will be written in
6258 square brackets behind each index value in the default and LaTeX output
6263 symbol x("x"), y("y");
6264 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
6265 ex e = indexed(x, mu) * indexed(y, nu);
6268 // prints 'x~mu*y~nu'
6269 cout << index_dimensions << e << endl;
6270 // prints 'x~mu[4]*y~nu[4]'
6271 cout << no_index_dimensions << e << endl;
6272 // prints 'x~mu*y~nu'
6277 @cindex Tree traversal
6278 If you need any fancy special output format, e.g. for interfacing GiNaC
6279 with other algebra systems or for producing code for different
6280 programming languages, you can always traverse the expression tree yourself:
6283 static void my_print(const ex & e)
6285 if (is_a<function>(e))
6286 cout << ex_to<function>(e).get_name();
6288 cout << ex_to<basic>(e).class_name();
6290 size_t n = e.nops();
6292 for (size_t i=0; i<n; i++) @{
6304 my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
6312 add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
6313 symbol(y))),numeric(-2)))
6316 If you need an output format that makes it possible to accurately
6317 reconstruct an expression by feeding the output to a suitable parser or
6318 object factory, you should consider storing the expression in an
6319 @code{archive} object and reading the object properties from there.
6320 See the section on archiving for more information.
6323 @subsection Expression input
6324 @cindex input of expressions
6326 GiNaC provides no way to directly read an expression from a stream because
6327 you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
6328 and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
6329 @code{y} you defined in your program and there is no way to specify the
6330 desired symbols to the @code{>>} stream input operator.
6332 Instead, GiNaC lets you construct an expression from a string, specifying the
6333 list of symbols to be used:
6337 symbol x("x"), y("y");
6338 ex e("2*x+sin(y)", lst(x, y));
6342 The input syntax is the same as that used by @command{ginsh} and the stream
6343 output operator @code{<<}. The symbols in the string are matched by name to
6344 the symbols in the list and if GiNaC encounters a symbol not specified in
6345 the list it will throw an exception.
6347 With this constructor, it's also easy to implement interactive GiNaC programs:
6352 #include <stdexcept>
6353 #include <ginac/ginac.h>
6354 using namespace std;
6355 using namespace GiNaC;
6362 cout << "Enter an expression containing 'x': ";
6367 cout << "The derivative of " << e << " with respect to x is ";
6368 cout << e.diff(x) << ".\n";
6369 @} catch (exception &p) @{
6370 cerr << p.what() << endl;
6376 @subsection Archiving
6377 @cindex @code{archive} (class)
6380 GiNaC allows creating @dfn{archives} of expressions which can be stored
6381 to or retrieved from files. To create an archive, you declare an object
6382 of class @code{archive} and archive expressions in it, giving each
6383 expression a unique name:
6387 using namespace std;
6388 #include <ginac/ginac.h>
6389 using namespace GiNaC;
6393 symbol x("x"), y("y"), z("z");
6395 ex foo = sin(x + 2*y) + 3*z + 41;
6399 a.archive_ex(foo, "foo");
6400 a.archive_ex(bar, "the second one");
6404 The archive can then be written to a file:
6408 ofstream out("foobar.gar");
6414 The file @file{foobar.gar} contains all information that is needed to
6415 reconstruct the expressions @code{foo} and @code{bar}.
6417 @cindex @command{viewgar}
6418 The tool @command{viewgar} that comes with GiNaC can be used to view
6419 the contents of GiNaC archive files:
6422 $ viewgar foobar.gar
6423 foo = 41+sin(x+2*y)+3*z
6424 the second one = 42+sin(x+2*y)+3*z
6427 The point of writing archive files is of course that they can later be
6433 ifstream in("foobar.gar");
6438 And the stored expressions can be retrieved by their name:
6445 ex ex1 = a2.unarchive_ex(syms, "foo");
6446 ex ex2 = a2.unarchive_ex(syms, "the second one");
6448 cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
6449 cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
6450 cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
6454 Note that you have to supply a list of the symbols which are to be inserted
6455 in the expressions. Symbols in archives are stored by their name only and
6456 if you don't specify which symbols you have, unarchiving the expression will
6457 create new symbols with that name. E.g. if you hadn't included @code{x} in
6458 the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
6459 have had no effect because the @code{x} in @code{ex1} would have been a
6460 different symbol than the @code{x} which was defined at the beginning of
6461 the program, although both would appear as @samp{x} when printed.
6463 You can also use the information stored in an @code{archive} object to
6464 output expressions in a format suitable for exact reconstruction. The
6465 @code{archive} and @code{archive_node} classes have a couple of member
6466 functions that let you access the stored properties:
6469 static void my_print2(const archive_node & n)
6472 n.find_string("class", class_name);
6473 cout << class_name << "(";
6475 archive_node::propinfovector p;
6476 n.get_properties(p);
6478 size_t num = p.size();
6479 for (size_t i=0; i<num; i++) @{
6480 const string &name = p[i].name;
6481 if (name == "class")
6483 cout << name << "=";
6485 unsigned count = p[i].count;
6489 for (unsigned j=0; j<count; j++) @{
6490 switch (p[i].type) @{
6491 case archive_node::PTYPE_BOOL: @{
6493 n.find_bool(name, x, j);
6494 cout << (x ? "true" : "false");
6497 case archive_node::PTYPE_UNSIGNED: @{
6499 n.find_unsigned(name, x, j);
6503 case archive_node::PTYPE_STRING: @{
6505 n.find_string(name, x, j);
6506 cout << '\"' << x << '\"';
6509 case archive_node::PTYPE_NODE: @{
6510 const archive_node &x = n.find_ex_node(name, j);
6532 ex e = pow(2, x) - y;
6534 my_print2(ar.get_top_node(0)); cout << endl;
6542 add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
6543 symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
6544 overall_coeff=numeric(number="0"))
6547 Be warned, however, that the set of properties and their meaning for each
6548 class may change between GiNaC versions.
6551 @node Extending GiNaC, What does not belong into GiNaC, Input/output, Top
6552 @c node-name, next, previous, up
6553 @chapter Extending GiNaC
6555 By reading so far you should have gotten a fairly good understanding of
6556 GiNaC's design patterns. From here on you should start reading the
6557 sources. All we can do now is issue some recommendations how to tackle
6558 GiNaC's many loose ends in order to fulfill everybody's dreams. If you
6559 develop some useful extension please don't hesitate to contact the GiNaC
6560 authors---they will happily incorporate them into future versions.
6563 * What does not belong into GiNaC:: What to avoid.
6564 * Symbolic functions:: Implementing symbolic functions.
6565 * Printing:: Adding new output formats.
6566 * Structures:: Defining new algebraic classes (the easy way).
6567 * Adding classes:: Defining new algebraic classes (the hard way).
6571 @node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
6572 @c node-name, next, previous, up
6573 @section What doesn't belong into GiNaC
6575 @cindex @command{ginsh}
6576 First of all, GiNaC's name must be read literally. It is designed to be
6577 a library for use within C++. The tiny @command{ginsh} accompanying
6578 GiNaC makes this even more clear: it doesn't even attempt to provide a
6579 language. There are no loops or conditional expressions in
6580 @command{ginsh}, it is merely a window into the library for the
6581 programmer to test stuff (or to show off). Still, the design of a
6582 complete CAS with a language of its own, graphical capabilities and all
6583 this on top of GiNaC is possible and is without doubt a nice project for
6586 There are many built-in functions in GiNaC that do not know how to
6587 evaluate themselves numerically to a precision declared at runtime
6588 (using @code{Digits}). Some may be evaluated at certain points, but not
6589 generally. This ought to be fixed. However, doing numerical
6590 computations with GiNaC's quite abstract classes is doomed to be
6591 inefficient. For this purpose, the underlying foundation classes
6592 provided by CLN are much better suited.
6595 @node Symbolic functions, Printing, What does not belong into GiNaC, Extending GiNaC
6596 @c node-name, next, previous, up
6597 @section Symbolic functions
6599 The easiest and most instructive way to start extending GiNaC is probably to
6600 create your own symbolic functions. These are implemented with the help of
6601 two preprocessor macros:
6603 @cindex @code{DECLARE_FUNCTION}
6604 @cindex @code{REGISTER_FUNCTION}
6606 DECLARE_FUNCTION_<n>P(<name>)
6607 REGISTER_FUNCTION(<name>, <options>)
6610 The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
6611 declares a C++ function with the given @samp{name} that takes exactly @samp{n}
6612 parameters of type @code{ex} and returns a newly constructed GiNaC
6613 @code{function} object that represents your function.
6615 The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
6616 the same @samp{name} as the respective @code{DECLARE_FUNCTION} macro, and a
6617 set of options that associate the symbolic function with C++ functions you
6618 provide to implement the various methods such as evaluation, derivative,
6619 series expansion etc. They also describe additional attributes the function
6620 might have, such as symmetry and commutation properties, and a name for
6621 LaTeX output. Multiple options are separated by the member access operator
6622 @samp{.} and can be given in an arbitrary order.
6624 (By the way: in case you are worrying about all the macros above we can
6625 assure you that functions are GiNaC's most macro-intense classes. We have
6626 done our best to avoid macros where we can.)
6628 @subsection A minimal example
6630 Here is an example for the implementation of a function with two arguments
6631 that is not further evaluated:
6634 DECLARE_FUNCTION_2P(myfcn)
6636 REGISTER_FUNCTION(myfcn, dummy())
6639 Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
6640 in algebraic expressions:
6646 ex e = 2*myfcn(42, 1+3*x) - x;
6648 // prints '2*myfcn(42,1+3*x)-x'
6653 The @code{dummy()} option in the @code{REGISTER_FUNCTION} line signifies
6654 "no options". A function with no options specified merely acts as a kind of
6655 container for its arguments. It is a pure "dummy" function with no associated
6656 logic (which is, however, sometimes perfectly sufficient).
6658 Let's now have a look at the implementation of GiNaC's cosine function for an
6659 example of how to make an "intelligent" function.
6661 @subsection The cosine function
6663 The GiNaC header file @file{inifcns.h} contains the line
6666 DECLARE_FUNCTION_1P(cos)
6669 which declares to all programs using GiNaC that there is a function @samp{cos}
6670 that takes one @code{ex} as an argument. This is all they need to know to use
6671 this function in expressions.
6673 The implementation of the cosine function is in @file{inifcns_trans.cpp}. Here
6674 is its @code{REGISTER_FUNCTION} line:
6677 REGISTER_FUNCTION(cos, eval_func(cos_eval).
6678 evalf_func(cos_evalf).
6679 derivative_func(cos_deriv).
6680 latex_name("\\cos"));
6683 There are four options defined for the cosine function. One of them
6684 (@code{latex_name}) gives the function a proper name for LaTeX output; the
6685 other three indicate the C++ functions in which the "brains" of the cosine
6686 function are defined.
6688 @cindex @code{hold()}
6690 The @code{eval_func()} option specifies the C++ function that implements
6691 the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
6692 the same number of arguments as the associated symbolic function (one in this
6693 case) and returns the (possibly transformed or in some way simplified)
6694 symbolically evaluated function (@xref{Automatic evaluation}, for a description
6695 of the automatic evaluation process). If no (further) evaluation is to take
6696 place, the @code{eval_func()} function must return the original function
6697 with @code{.hold()}, to avoid a potential infinite recursion. If your
6698 symbolic functions produce a segmentation fault or stack overflow when
6699 using them in expressions, you are probably missing a @code{.hold()}
6702 The @code{eval_func()} function for the cosine looks something like this
6703 (actually, it doesn't look like this at all, but it should give you an idea
6707 static ex cos_eval(const ex & x)
6709 if ("x is a multiple of 2*Pi")
6711 else if ("x is a multiple of Pi")
6713 else if ("x is a multiple of Pi/2")
6717 else if ("x has the form 'acos(y)'")
6719 else if ("x has the form 'asin(y)'")
6724 return cos(x).hold();
6728 This function is called every time the cosine is used in a symbolic expression:
6734 // this calls cos_eval(Pi), and inserts its return value into
6735 // the actual expression
6742 In this way, @code{cos(4*Pi)} automatically becomes @math{1},
6743 @code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
6744 symbolic transformation can be done, the unmodified function is returned
6745 with @code{.hold()}.
6747 GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
6748 The user has to call @code{evalf()} for that. This is implemented in a
6752 static ex cos_evalf(const ex & x)
6754 if (is_a<numeric>(x))
6755 return cos(ex_to<numeric>(x));
6757 return cos(x).hold();
6761 Since we are lazy we defer the problem of numeric evaluation to somebody else,
6762 in this case the @code{cos()} function for @code{numeric} objects, which in
6763 turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
6764 isn't really needed here, but reminds us that the corresponding @code{eval()}
6765 function would require it in this place.
6767 Differentiation will surely turn up and so we need to tell @code{cos}
6768 what its first derivative is (higher derivatives, @code{.diff(x,3)} for
6769 instance, are then handled automatically by @code{basic::diff} and
6773 static ex cos_deriv(const ex & x, unsigned diff_param)
6779 @cindex product rule
6780 The second parameter is obligatory but uninteresting at this point. It
6781 specifies which parameter to differentiate in a partial derivative in
6782 case the function has more than one parameter, and its main application
6783 is for correct handling of the chain rule.
6785 An implementation of the series expansion is not needed for @code{cos()} as
6786 it doesn't have any poles and GiNaC can do Taylor expansion by itself (as
6787 long as it knows what the derivative of @code{cos()} is). @code{tan()}, on
6788 the other hand, does have poles and may need to do Laurent expansion:
6791 static ex tan_series(const ex & x, const relational & rel,
6792 int order, unsigned options)
6794 // Find the actual expansion point
6795 const ex x_pt = x.subs(rel);
6797 if ("x_pt is not an odd multiple of Pi/2")
6798 throw do_taylor(); // tell function::series() to do Taylor expansion
6800 // On a pole, expand sin()/cos()
6801 return (sin(x)/cos(x)).series(rel, order+2, options);
6805 The @code{series()} implementation of a function @emph{must} return a
6806 @code{pseries} object, otherwise your code will crash.
6808 @subsection Function options
6810 GiNaC functions understand several more options which are always
6811 specified as @code{.option(params)}. None of them are required, but you
6812 need to specify at least one option to @code{REGISTER_FUNCTION()}. There
6813 is a do-nothing option called @code{dummy()} which you can use to define
6814 functions without any special options.
6817 eval_func(<C++ function>)
6818 evalf_func(<C++ function>)
6819 derivative_func(<C++ function>)
6820 series_func(<C++ function>)
6821 conjugate_func(<C++ function>)
6824 These specify the C++ functions that implement symbolic evaluation,
6825 numeric evaluation, partial derivatives, and series expansion, respectively.
6826 They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
6827 @code{diff()} and @code{series()}.
6829 The @code{eval_func()} function needs to use @code{.hold()} if no further
6830 automatic evaluation is desired or possible.
6832 If no @code{series_func()} is given, GiNaC defaults to simple Taylor
6833 expansion, which is correct if there are no poles involved. If the function
6834 has poles in the complex plane, the @code{series_func()} needs to check
6835 whether the expansion point is on a pole and fall back to Taylor expansion
6836 if it isn't. Otherwise, the pole usually needs to be regularized by some
6837 suitable transformation.
6840 latex_name(const string & n)
6843 specifies the LaTeX code that represents the name of the function in LaTeX
6844 output. The default is to put the function name in an @code{\mbox@{@}}.
6847 do_not_evalf_params()
6850 This tells @code{evalf()} to not recursively evaluate the parameters of the
6851 function before calling the @code{evalf_func()}.
6854 set_return_type(unsigned return_type, unsigned return_type_tinfo)
6857 This allows you to explicitly specify the commutation properties of the
6858 function (@xref{Non-commutative objects}, for an explanation of
6859 (non)commutativity in GiNaC). For example, you can use
6860 @code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
6861 GiNaC treat your function like a matrix. By default, functions inherit the
6862 commutation properties of their first argument.
6865 set_symmetry(const symmetry & s)
6868 specifies the symmetry properties of the function with respect to its
6869 arguments. @xref{Indexed objects}, for an explanation of symmetry
6870 specifications. GiNaC will automatically rearrange the arguments of
6871 symmetric functions into a canonical order.
6873 Sometimes you may want to have finer control over how functions are
6874 displayed in the output. For example, the @code{abs()} function prints
6875 itself as @samp{abs(x)} in the default output format, but as @samp{|x|}
6876 in LaTeX mode, and @code{fabs(x)} in C source output. This is achieved
6880 print_func<C>(<C++ function>)
6883 option which is explained in the next section.
6885 @subsection Functions with a variable number of arguments
6887 The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
6888 functions with a fixed number of arguments. Sometimes, though, you may need
6889 to have a function that accepts a variable number of expressions. One way to
6890 accomplish this is to pass variable-length lists as arguments. The
6891 @code{Li()} function uses this method for multiple polylogarithms.
6893 It is also possible to define functions that accept a different number of
6894 parameters under the same function name, such as the @code{psi()} function
6895 which can be called either as @code{psi(z)} (the digamma function) or as
6896 @code{psi(n, z)} (polygamma functions). These are actually two different
6897 functions in GiNaC that, however, have the same name. Defining such
6898 functions is not possible with the macros but requires manually fiddling
6899 with GiNaC internals. If you are interested, please consult the GiNaC source
6900 code for the @code{psi()} function (@file{inifcns.h} and
6901 @file{inifcns_gamma.cpp}).
6904 @node Printing, Structures, Symbolic functions, Extending GiNaC
6905 @c node-name, next, previous, up
6906 @section GiNaC's expression output system
6908 GiNaC allows the output of expressions in a variety of different formats
6909 (@pxref{Input/output}). This section will explain how expression output
6910 is implemented internally, and how to define your own output formats or
6911 change the output format of built-in algebraic objects. You will also want
6912 to read this section if you plan to write your own algebraic classes or
6915 @cindex @code{print_context} (class)
6916 @cindex @code{print_dflt} (class)
6917 @cindex @code{print_latex} (class)
6918 @cindex @code{print_tree} (class)
6919 @cindex @code{print_csrc} (class)
6920 All the different output formats are represented by a hierarchy of classes
6921 rooted in the @code{print_context} class, defined in the @file{print.h}
6926 the default output format
6928 output in LaTeX mathematical mode
6930 a dump of the internal expression structure (for debugging)
6932 the base class for C source output
6933 @item print_csrc_float
6934 C source output using the @code{float} type
6935 @item print_csrc_double
6936 C source output using the @code{double} type
6937 @item print_csrc_cl_N
6938 C source output using CLN types
6941 The @code{print_context} base class provides two public data members:
6953 @code{s} is a reference to the stream to output to, while @code{options}
6954 holds flags and modifiers. Currently, there is only one flag defined:
6955 @code{print_options::print_index_dimensions} instructs the @code{idx} class
6956 to print the index dimension which is normally hidden.
6958 When you write something like @code{std::cout << e}, where @code{e} is
6959 an object of class @code{ex}, GiNaC will construct an appropriate
6960 @code{print_context} object (of a class depending on the selected output
6961 format), fill in the @code{s} and @code{options} members, and call
6963 @cindex @code{print()}
6965 void ex::print(const print_context & c, unsigned level = 0) const;
6968 which in turn forwards the call to the @code{print()} method of the
6969 top-level algebraic object contained in the expression.
6971 Unlike other methods, GiNaC classes don't usually override their
6972 @code{print()} method to implement expression output. Instead, the default
6973 implementation @code{basic::print(c, level)} performs a run-time double
6974 dispatch to a function selected by the dynamic type of the object and the
6975 passed @code{print_context}. To this end, GiNaC maintains a separate method
6976 table for each class, similar to the virtual function table used for ordinary
6977 (single) virtual function dispatch.
6979 The method table contains one slot for each possible @code{print_context}
6980 type, indexed by the (internally assigned) serial number of the type. Slots
6981 may be empty, in which case GiNaC will retry the method lookup with the
6982 @code{print_context} object's parent class, possibly repeating the process
6983 until it reaches the @code{print_context} base class. If there's still no
6984 method defined, the method table of the algebraic object's parent class
6985 is consulted, and so on, until a matching method is found (eventually it
6986 will reach the combination @code{basic/print_context}, which prints the
6987 object's class name enclosed in square brackets).
6989 You can think of the print methods of all the different classes and output
6990 formats as being arranged in a two-dimensional matrix with one axis listing
6991 the algebraic classes and the other axis listing the @code{print_context}
6994 Subclasses of @code{basic} can, of course, also overload @code{basic::print()}
6995 to implement printing, but then they won't get any of the benefits of the
6996 double dispatch mechanism (such as the ability for derived classes to
6997 inherit only certain print methods from its parent, or the replacement of
6998 methods at run-time).
7000 @subsection Print methods for classes
7002 The method table for a class is set up either in the definition of the class,
7003 by passing the appropriate @code{print_func<C>()} option to
7004 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()} (@xref{Adding classes}, for
7005 an example), or at run-time using @code{set_print_func<T, C>()}. The latter
7006 can also be used to override existing methods dynamically.
7008 The argument to @code{print_func<C>()} and @code{set_print_func<T, C>()} can
7009 be a member function of the class (or one of its parent classes), a static
7010 member function, or an ordinary (global) C++ function. The @code{C} template
7011 parameter specifies the appropriate @code{print_context} type for which the
7012 method should be invoked, while, in the case of @code{set_print_func<>()}, the
7013 @code{T} parameter specifies the algebraic class (for @code{print_func<>()},
7014 the class is the one being implemented by
7015 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}).
7017 For print methods that are member functions, their first argument must be of
7018 a type convertible to a @code{const C &}, and the second argument must be an
7021 For static members and global functions, the first argument must be of a type
7022 convertible to a @code{const T &}, the second argument must be of a type
7023 convertible to a @code{const C &}, and the third argument must be an
7024 @code{unsigned}. A global function will, of course, not have access to
7025 private and protected members of @code{T}.
7027 The @code{unsigned} argument of the print methods (and of @code{ex::print()}
7028 and @code{basic::print()}) is used for proper parenthesizing of the output
7029 (and by @code{print_tree} for proper indentation). It can be used for similar
7030 purposes if you write your own output formats.
7032 The explanations given above may seem complicated, but in practice it's
7033 really simple, as shown in the following example. Suppose that we want to
7034 display exponents in LaTeX output not as superscripts but with little
7035 upwards-pointing arrows. This can be achieved in the following way:
7038 void my_print_power_as_latex(const power & p,
7039 const print_latex & c,
7042 // get the precedence of the 'power' class
7043 unsigned power_prec = p.precedence();
7045 // if the parent operator has the same or a higher precedence
7046 // we need parentheses around the power
7047 if (level >= power_prec)
7050 // print the basis and exponent, each enclosed in braces, and
7051 // separated by an uparrow
7053 p.op(0).print(c, power_prec);
7054 c.s << "@}\\uparrow@{";
7055 p.op(1).print(c, power_prec);
7058 // don't forget the closing parenthesis
7059 if (level >= power_prec)
7065 // a sample expression
7066 symbol x("x"), y("y");
7067 ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;
7069 // switch to LaTeX mode
7072 // this prints "-1+@{(y+x)@}^@{2@}-3 \frac@{x^@{3@}@}@{y^@{2@}@}"
7075 // now we replace the method for the LaTeX output of powers with
7077 set_print_func<power, print_latex>(my_print_power_as_latex);
7079 // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
7090 The first argument of @code{my_print_power_as_latex} could also have been
7091 a @code{const basic &}, the second one a @code{const print_context &}.
7094 The above code depends on @code{mul} objects converting their operands to
7095 @code{power} objects for the purpose of printing.
7098 The output of products including negative powers as fractions is also
7099 controlled by the @code{mul} class.
7102 The @code{power/print_latex} method provided by GiNaC prints square roots
7103 using @code{\sqrt}, but the above code doesn't.
7107 It's not possible to restore a method table entry to its previous or default
7108 value. Once you have called @code{set_print_func()}, you can only override
7109 it with another call to @code{set_print_func()}, but you can't easily go back
7110 to the default behavior again (you can, of course, dig around in the GiNaC
7111 sources, find the method that is installed at startup
7112 (@code{power::do_print_latex} in this case), and @code{set_print_func} that
7113 one; that is, after you circumvent the C++ member access control@dots{}).
7115 @subsection Print methods for functions
7117 Symbolic functions employ a print method dispatch mechanism similar to the
7118 one used for classes. The methods are specified with @code{print_func<C>()}
7119 function options. If you don't specify any special print methods, the function
7120 will be printed with its name (or LaTeX name, if supplied), followed by a
7121 comma-separated list of arguments enclosed in parentheses.
7123 For example, this is what GiNaC's @samp{abs()} function is defined like:
7126 static ex abs_eval(const ex & arg) @{ ... @}
7127 static ex abs_evalf(const ex & arg) @{ ... @}
7129 static void abs_print_latex(const ex & arg, const print_context & c)
7131 c.s << "@{|"; arg.print(c); c.s << "|@}";
7134 static void abs_print_csrc_float(const ex & arg, const print_context & c)
7136 c.s << "fabs("; arg.print(c); c.s << ")";
7139 REGISTER_FUNCTION(abs, eval_func(abs_eval).
7140 evalf_func(abs_evalf).
7141 print_func<print_latex>(abs_print_latex).
7142 print_func<print_csrc_float>(abs_print_csrc_float).
7143 print_func<print_csrc_double>(abs_print_csrc_float));
7146 This will display @samp{abs(x)} as @samp{|x|} in LaTeX mode and @code{fabs(x)}
7147 in non-CLN C source output, but as @code{abs(x)} in all other formats.
7149 There is currently no equivalent of @code{set_print_func()} for functions.
7151 @subsection Adding new output formats
7153 Creating a new output format involves subclassing @code{print_context},
7154 which is somewhat similar to adding a new algebraic class
7155 (@pxref{Adding classes}). There is a macro @code{GINAC_DECLARE_PRINT_CONTEXT}
7156 that needs to go into the class definition, and a corresponding macro
7157 @code{GINAC_IMPLEMENT_PRINT_CONTEXT} that has to appear at global scope.
7158 Every @code{print_context} class needs to provide a default constructor
7159 and a constructor from an @code{std::ostream} and an @code{unsigned}
7162 Here is an example for a user-defined @code{print_context} class:
7165 class print_myformat : public print_dflt
7167 GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
7169 print_myformat(std::ostream & os, unsigned opt = 0)
7170 : print_dflt(os, opt) @{@}
7173 print_myformat::print_myformat() : print_dflt(std::cout) @{@}
7175 GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)
7178 That's all there is to it. None of the actual expression output logic is
7179 implemented in this class. It merely serves as a selector for choosing
7180 a particular format. The algorithms for printing expressions in the new
7181 format are implemented as print methods, as described above.
7183 @code{print_myformat} is a subclass of @code{print_dflt}, so it behaves
7184 exactly like GiNaC's default output format:
7189 ex e = pow(x, 2) + 1;
7191 // this prints "1+x^2"
7194 // this also prints "1+x^2"
7195 e.print(print_myformat()); cout << endl;
7201 To fill @code{print_myformat} with life, we need to supply appropriate
7202 print methods with @code{set_print_func()}, like this:
7205 // This prints powers with '**' instead of '^'. See the LaTeX output
7206 // example above for explanations.
7207 void print_power_as_myformat(const power & p,
7208 const print_myformat & c,
7211 unsigned power_prec = p.precedence();
7212 if (level >= power_prec)
7214 p.op(0).print(c, power_prec);
7216 p.op(1).print(c, power_prec);
7217 if (level >= power_prec)
7223 // install a new print method for power objects
7224 set_print_func<power, print_myformat>(print_power_as_myformat);
7226 // now this prints "1+x**2"
7227 e.print(print_myformat()); cout << endl;
7229 // but the default format is still "1+x^2"
7235 @node Structures, Adding classes, Printing, Extending GiNaC
7236 @c node-name, next, previous, up
7239 If you are doing some very specialized things with GiNaC, or if you just
7240 need some more organized way to store data in your expressions instead of
7241 anonymous lists, you may want to implement your own algebraic classes.
7242 ('algebraic class' means any class directly or indirectly derived from
7243 @code{basic} that can be used in GiNaC expressions).
7245 GiNaC offers two ways of accomplishing this: either by using the
7246 @code{structure<T>} template class, or by rolling your own class from
7247 scratch. This section will discuss the @code{structure<T>} template which
7248 is easier to use but more limited, while the implementation of custom
7249 GiNaC classes is the topic of the next section. However, you may want to
7250 read both sections because many common concepts and member functions are
7251 shared by both concepts, and it will also allow you to decide which approach
7252 is most suited to your needs.
7254 The @code{structure<T>} template, defined in the GiNaC header file
7255 @file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
7256 or @code{class}) into a GiNaC object that can be used in expressions.
7258 @subsection Example: scalar products
7260 Let's suppose that we need a way to handle some kind of abstract scalar
7261 product of the form @samp{<x|y>} in expressions. Objects of the scalar
7262 product class have to store their left and right operands, which can in turn
7263 be arbitrary expressions. Here is a possible way to represent such a
7264 product in a C++ @code{struct}:
7268 using namespace std;
7270 #include <ginac/ginac.h>
7271 using namespace GiNaC;
7277 sprod_s(ex l, ex r) : left(l), right(r) @{@}
7281 The default constructor is required. Now, to make a GiNaC class out of this
7282 data structure, we need only one line:
7285 typedef structure<sprod_s> sprod;
7288 That's it. This line constructs an algebraic class @code{sprod} which
7289 contains objects of type @code{sprod_s}. We can now use @code{sprod} in
7290 expressions like any other GiNaC class:
7294 symbol a("a"), b("b");
7295 ex e = sprod(sprod_s(a, b));
7299 Note the difference between @code{sprod} which is the algebraic class, and
7300 @code{sprod_s} which is the unadorned C++ structure containing the @code{left}
7301 and @code{right} data members. As shown above, an @code{sprod} can be
7302 constructed from an @code{sprod_s} object.
7304 If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
7305 you could define a little wrapper function like this:
7308 inline ex make_sprod(ex left, ex right)
7310 return sprod(sprod_s(left, right));
7314 The @code{sprod_s} object contained in @code{sprod} can be accessed with
7315 the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
7316 @code{get_struct()}:
7320 cout << ex_to<sprod>(e)->left << endl;
7322 cout << ex_to<sprod>(e).get_struct().right << endl;
7327 You only have read access to the members of @code{sprod_s}.
7329 The type definition of @code{sprod} is enough to write your own algorithms
7330 that deal with scalar products, for example:
7335 if (is_a<sprod>(p)) @{
7336 const sprod_s & sp = ex_to<sprod>(p).get_struct();
7337 return make_sprod(sp.right, sp.left);
7348 @subsection Structure output
7350 While the @code{sprod} type is useable it still leaves something to be
7351 desired, most notably proper output:
7356 // -> [structure object]
7360 By default, any structure types you define will be printed as
7361 @samp{[structure object]}. To override this you can either specialize the
7362 template's @code{print()} member function, or specify print methods with
7363 @code{set_print_func<>()}, as described in @ref{Printing}. Unfortunately,
7364 it's not possible to supply class options like @code{print_func<>()} to
7365 structures, so for a self-contained structure type you need to resort to
7366 overriding the @code{print()} function, which is also what we will do here.
7368 The member functions of GiNaC classes are described in more detail in the
7369 next section, but it shouldn't be hard to figure out what's going on here:
7372 void sprod::print(const print_context & c, unsigned level) const
7374 // tree debug output handled by superclass
7375 if (is_a<print_tree>(c))
7376 inherited::print(c, level);
7378 // get the contained sprod_s object
7379 const sprod_s & sp = get_struct();
7381 // print_context::s is a reference to an ostream
7382 c.s << "<" << sp.left << "|" << sp.right << ">";
7386 Now we can print expressions containing scalar products:
7392 cout << swap_sprod(e) << endl;
7397 @subsection Comparing structures
7399 The @code{sprod} class defined so far still has one important drawback: all
7400 scalar products are treated as being equal because GiNaC doesn't know how to
7401 compare objects of type @code{sprod_s}. This can lead to some confusing
7402 and undesired behavior:
7406 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7408 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7409 // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
7413 To remedy this, we first need to define the operators @code{==} and @code{<}
7414 for objects of type @code{sprod_s}:
7417 inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
7419 return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
7422 inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
7424 return lhs.left.compare(rhs.left) < 0
7425 ? true : lhs.right.compare(rhs.right) < 0;
7429 The ordering established by the @code{<} operator doesn't have to make any
7430 algebraic sense, but it needs to be well defined. Note that we can't use
7431 expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
7432 in the implementation of these operators because they would construct
7433 GiNaC @code{relational} objects which in the case of @code{<} do not
7434 establish a well defined ordering (for arbitrary expressions, GiNaC can't
7435 decide which one is algebraically 'less').
7437 Next, we need to change our definition of the @code{sprod} type to let
7438 GiNaC know that an ordering relation exists for the embedded objects:
7441 typedef structure<sprod_s, compare_std_less> sprod;
7444 @code{sprod} objects then behave as expected:
7448 cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
7449 // -> <a|b>-<a^2|b^2>
7450 cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
7451 // -> <a|b>+<a^2|b^2>
7452 cout << make_sprod(a, b) - make_sprod(a, b) << endl;
7454 cout << make_sprod(a, b) + make_sprod(a, b) << endl;
7459 The @code{compare_std_less} policy parameter tells GiNaC to use the
7460 @code{std::less} and @code{std::equal_to} functors to compare objects of
7461 type @code{sprod_s}. By default, these functors forward their work to the
7462 standard @code{<} and @code{==} operators, which we have overloaded.
7463 Alternatively, we could have specialized @code{std::less} and
7464 @code{std::equal_to} for class @code{sprod_s}.
7466 GiNaC provides two other comparison policies for @code{structure<T>}
7467 objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
7468 which does a bit-wise comparison of the contained @code{T} objects.
7469 This should be used with extreme care because it only works reliably with
7470 built-in integral types, and it also compares any padding (filler bytes of
7471 undefined value) that the @code{T} class might have.
7473 @subsection Subexpressions
7475 Our scalar product class has two subexpressions: the left and right
7476 operands. It might be a good idea to make them accessible via the standard
7477 @code{nops()} and @code{op()} methods:
7480 size_t sprod::nops() const
7485 ex sprod::op(size_t i) const
7489 return get_struct().left;
7491 return get_struct().right;
7493 throw std::range_error("sprod::op(): no such operand");
7498 Implementing @code{nops()} and @code{op()} for container types such as
7499 @code{sprod} has two other nice side effects:
7503 @code{has()} works as expected
7505 GiNaC generates better hash keys for the objects (the default implementation
7506 of @code{calchash()} takes subexpressions into account)
7509 @cindex @code{let_op()}
7510 There is a non-const variant of @code{op()} called @code{let_op()} that
7511 allows replacing subexpressions:
7514 ex & sprod::let_op(size_t i)
7516 // every non-const member function must call this
7517 ensure_if_modifiable();
7521 return get_struct().left;
7523 return get_struct().right;
7525 throw std::range_error("sprod::let_op(): no such operand");
7530 Once we have provided @code{let_op()} we also get @code{subs()} and
7531 @code{map()} for free. In fact, every container class that returns a non-null
7532 @code{nops()} value must either implement @code{let_op()} or provide custom
7533 implementations of @code{subs()} and @code{map()}.
7535 In turn, the availability of @code{map()} enables the recursive behavior of a
7536 couple of other default method implementations, in particular @code{evalf()},
7537 @code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
7538 we probably want to provide our own version of @code{expand()} for scalar
7539 products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
7540 This is left as an exercise for the reader.
7542 The @code{structure<T>} template defines many more member functions that
7543 you can override by specialization to customize the behavior of your
7544 structures. You are referred to the next section for a description of
7545 some of these (especially @code{eval()}). There is, however, one topic
7546 that shall be addressed here, as it demonstrates one peculiarity of the
7547 @code{structure<T>} template: archiving.
7549 @subsection Archiving structures
7551 If you don't know how the archiving of GiNaC objects is implemented, you
7552 should first read the next section and then come back here. You're back?
7555 To implement archiving for structures it is not enough to provide
7556 specializations for the @code{archive()} member function and the
7557 unarchiving constructor (the @code{unarchive()} function has a default
7558 implementation). You also need to provide a unique name (as a string literal)
7559 for each structure type you define. This is because in GiNaC archives,
7560 the class of an object is stored as a string, the class name.
7562 By default, this class name (as returned by the @code{class_name()} member
7563 function) is @samp{structure} for all structure classes. This works as long
7564 as you have only defined one structure type, but if you use two or more you
7565 need to provide a different name for each by specializing the
7566 @code{get_class_name()} member function. Here is a sample implementation
7567 for enabling archiving of the scalar product type defined above:
7570 const char *sprod::get_class_name() @{ return "sprod"; @}
7572 void sprod::archive(archive_node & n) const
7574 inherited::archive(n);
7575 n.add_ex("left", get_struct().left);
7576 n.add_ex("right", get_struct().right);
7579 sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
7581 n.find_ex("left", get_struct().left, sym_lst);
7582 n.find_ex("right", get_struct().right, sym_lst);
7586 Note that the unarchiving constructor is @code{sprod::structure} and not
7587 @code{sprod::sprod}, and that we don't need to supply an
7588 @code{sprod::unarchive()} function.
7591 @node Adding classes, A comparison with other CAS, Structures, Extending GiNaC
7592 @c node-name, next, previous, up
7593 @section Adding classes
7595 The @code{structure<T>} template provides an way to extend GiNaC with custom
7596 algebraic classes that is easy to use but has its limitations, the most
7597 severe of which being that you can't add any new member functions to
7598 structures. To be able to do this, you need to write a new class definition
7601 This section will explain how to implement new algebraic classes in GiNaC by
7602 giving the example of a simple 'string' class. After reading this section
7603 you will know how to properly declare a GiNaC class and what the minimum
7604 required member functions are that you have to implement. We only cover the
7605 implementation of a 'leaf' class here (i.e. one that doesn't contain
7606 subexpressions). Creating a container class like, for example, a class
7607 representing tensor products is more involved but this section should give
7608 you enough information so you can consult the source to GiNaC's predefined
7609 classes if you want to implement something more complicated.
7611 @subsection GiNaC's run-time type information system
7613 @cindex hierarchy of classes
7615 All algebraic classes (that is, all classes that can appear in expressions)
7616 in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
7617 @code{basic *} (which is essentially what an @code{ex} is) represents a
7618 generic pointer to an algebraic class. Occasionally it is necessary to find
7619 out what the class of an object pointed to by a @code{basic *} really is.
7620 Also, for the unarchiving of expressions it must be possible to find the
7621 @code{unarchive()} function of a class given the class name (as a string). A
7622 system that provides this kind of information is called a run-time type
7623 information (RTTI) system. The C++ language provides such a thing (see the
7624 standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
7625 implements its own, simpler RTTI.
7627 The RTTI in GiNaC is based on two mechanisms:
7632 The @code{basic} class declares a member variable @code{tinfo_key} which
7633 holds an unsigned integer that identifies the object's class. These numbers
7634 are defined in the @file{tinfos.h} header file for the built-in GiNaC
7635 classes. They all start with @code{TINFO_}.
7638 By means of some clever tricks with static members, GiNaC maintains a list
7639 of information for all classes derived from @code{basic}. The information
7640 available includes the class names, the @code{tinfo_key}s, and pointers
7641 to the unarchiving functions. This class registry is defined in the
7642 @file{registrar.h} header file.
7646 The disadvantage of this proprietary RTTI implementation is that there's
7647 a little more to do when implementing new classes (C++'s RTTI works more
7648 or less automatically) but don't worry, most of the work is simplified by
7651 @subsection A minimalistic example
7653 Now we will start implementing a new class @code{mystring} that allows
7654 placing character strings in algebraic expressions (this is not very useful,
7655 but it's just an example). This class will be a direct subclass of
7656 @code{basic}. You can use this sample implementation as a starting point
7657 for your own classes.
7659 The code snippets given here assume that you have included some header files
7665 #include <stdexcept>
7666 using namespace std;
7668 #include <ginac/ginac.h>
7669 using namespace GiNaC;
7672 The first thing we have to do is to define a @code{tinfo_key} for our new
7673 class. This can be any arbitrary unsigned number that is not already taken
7674 by one of the existing classes but it's better to come up with something
7675 that is unlikely to clash with keys that might be added in the future. The
7676 numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
7677 which is not a requirement but we are going to stick with this scheme:
7680 const unsigned TINFO_mystring = 0x42420001U;
7683 Now we can write down the class declaration. The class stores a C++
7684 @code{string} and the user shall be able to construct a @code{mystring}
7685 object from a C or C++ string:
7688 class mystring : public basic
7690 GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
7693 mystring(const string &s);
7694 mystring(const char *s);
7700 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7703 The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
7704 macros are defined in @file{registrar.h}. They take the name of the class
7705 and its direct superclass as arguments and insert all required declarations
7706 for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
7707 the first line after the opening brace of the class definition. The
7708 @code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
7709 source (at global scope, of course, not inside a function).
7711 @code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
7712 declarations of the default constructor and a couple of other functions that
7713 are required. It also defines a type @code{inherited} which refers to the
7714 superclass so you don't have to modify your code every time you shuffle around
7715 the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
7716 class with the GiNaC RTTI (there is also a
7717 @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
7718 options for the class, and which we will be using instead in a few minutes).
7720 Now there are seven member functions we have to implement to get a working
7726 @code{mystring()}, the default constructor.
7729 @code{void archive(archive_node &n)}, the archiving function. This stores all
7730 information needed to reconstruct an object of this class inside an
7731 @code{archive_node}.
7734 @code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
7735 constructor. This constructs an instance of the class from the information
7736 found in an @code{archive_node}.
7739 @code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
7740 unarchiving function. It constructs a new instance by calling the unarchiving
7744 @cindex @code{compare_same_type()}
7745 @code{int compare_same_type(const basic &other)}, which is used internally
7746 by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
7747 -1, depending on the relative order of this object and the @code{other}
7748 object. If it returns 0, the objects are considered equal.
7749 @strong{Please notice:} This has nothing to do with the (numeric) ordering
7750 relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
7751 for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
7752 may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
7753 must provide a @code{compare_same_type()} function, even those representing
7754 objects for which no reasonable algebraic ordering relationship can be
7758 And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
7759 which are the two constructors we declared.
7763 Let's proceed step-by-step. The default constructor looks like this:
7766 mystring::mystring() : inherited(TINFO_mystring) @{@}
7769 The golden rule is that in all constructors you have to set the
7770 @code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
7771 it will be set by the constructor of the superclass and all hell will break
7772 loose in the RTTI. For your convenience, the @code{basic} class provides
7773 a constructor that takes a @code{tinfo_key} value, which we are using here
7774 (remember that in our case @code{inherited == basic}). If the superclass
7775 didn't have such a constructor, we would have to set the @code{tinfo_key}
7776 to the right value manually.
7778 In the default constructor you should set all other member variables to
7779 reasonable default values (we don't need that here since our @code{str}
7780 member gets set to an empty string automatically).
7782 Next are the three functions for archiving. You have to implement them even
7783 if you don't plan to use archives, but the minimum required implementation
7784 is really simple. First, the archiving function:
7787 void mystring::archive(archive_node &n) const
7789 inherited::archive(n);
7790 n.add_string("string", str);
7794 The only thing that is really required is calling the @code{archive()}
7795 function of the superclass. Optionally, you can store all information you
7796 deem necessary for representing the object into the passed
7797 @code{archive_node}. We are just storing our string here. For more
7798 information on how the archiving works, consult the @file{archive.h} header
7801 The unarchiving constructor is basically the inverse of the archiving
7805 mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
7807 n.find_string("string", str);
7811 If you don't need archiving, just leave this function empty (but you must
7812 invoke the unarchiving constructor of the superclass). Note that we don't
7813 have to set the @code{tinfo_key} here because it is done automatically
7814 by the unarchiving constructor of the @code{basic} class.
7816 Finally, the unarchiving function:
7819 ex mystring::unarchive(const archive_node &n, lst &sym_lst)
7821 return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
7825 You don't have to understand how exactly this works. Just copy these
7826 four lines into your code literally (replacing the class name, of
7827 course). It calls the unarchiving constructor of the class and unless
7828 you are doing something very special (like matching @code{archive_node}s
7829 to global objects) you don't need a different implementation. For those
7830 who are interested: setting the @code{dynallocated} flag puts the object
7831 under the control of GiNaC's garbage collection. It will get deleted
7832 automatically once it is no longer referenced.
7834 Our @code{compare_same_type()} function uses a provided function to compare
7838 int mystring::compare_same_type(const basic &other) const
7840 const mystring &o = static_cast<const mystring &>(other);
7841 int cmpval = str.compare(o.str);
7844 else if (cmpval < 0)
7851 Although this function takes a @code{basic &}, it will always be a reference
7852 to an object of exactly the same class (objects of different classes are not
7853 comparable), so the cast is safe. If this function returns 0, the two objects
7854 are considered equal (in the sense that @math{A-B=0}), so you should compare
7855 all relevant member variables.
7857 Now the only thing missing is our two new constructors:
7860 mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
7861 mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
7864 No surprises here. We set the @code{str} member from the argument and
7865 remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
7867 That's it! We now have a minimal working GiNaC class that can store
7868 strings in algebraic expressions. Let's confirm that the RTTI works:
7871 ex e = mystring("Hello, world!");
7872 cout << is_a<mystring>(e) << endl;
7875 cout << ex_to<basic>(e).class_name() << endl;
7879 Obviously it does. Let's see what the expression @code{e} looks like:
7883 // -> [mystring object]
7886 Hm, not exactly what we expect, but of course the @code{mystring} class
7887 doesn't yet know how to print itself. This can be done either by implementing
7888 the @code{print()} member function, or, preferably, by specifying a
7889 @code{print_func<>()} class option. Let's say that we want to print the string
7890 surrounded by double quotes:
7893 class mystring : public basic
7897 void do_print(const print_context &c, unsigned level = 0) const;
7901 void mystring::do_print(const print_context &c, unsigned level) const
7903 // print_context::s is a reference to an ostream
7904 c.s << '\"' << str << '\"';
7908 The @code{level} argument is only required for container classes to
7909 correctly parenthesize the output.
7911 Now we need to tell GiNaC that @code{mystring} objects should use the
7912 @code{do_print()} member function for printing themselves. For this, we
7916 GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
7922 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
7923 print_func<print_context>(&mystring::do_print))
7926 Let's try again to print the expression:
7930 // -> "Hello, world!"
7933 Much better. If we wanted to have @code{mystring} objects displayed in a
7934 different way depending on the output format (default, LaTeX, etc.), we
7935 would have supplied multiple @code{print_func<>()} options with different
7936 template parameters (@code{print_dflt}, @code{print_latex}, etc.),
7937 separated by dots. This is similar to the way options are specified for
7938 symbolic functions. @xref{Printing}, for a more in-depth description of the
7939 way expression output is implemented in GiNaC.
7941 The @code{mystring} class can be used in arbitrary expressions:
7944 e += mystring("GiNaC rulez");
7946 // -> "GiNaC rulez"+"Hello, world!"
7949 (GiNaC's automatic term reordering is in effect here), or even
7952 e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
7954 // -> "One string"^(2*sin(-"Another string"+Pi))
7957 Whether this makes sense is debatable but remember that this is only an
7958 example. At least it allows you to implement your own symbolic algorithms
7961 Note that GiNaC's algebraic rules remain unchanged:
7964 e = mystring("Wow") * mystring("Wow");
7968 e = pow(mystring("First")-mystring("Second"), 2);
7969 cout << e.expand() << endl;
7970 // -> -2*"First"*"Second"+"First"^2+"Second"^2
7973 There's no way to, for example, make GiNaC's @code{add} class perform string
7974 concatenation. You would have to implement this yourself.
7976 @subsection Automatic evaluation
7979 @cindex @code{eval()}
7980 @cindex @code{hold()}
7981 When dealing with objects that are just a little more complicated than the
7982 simple string objects we have implemented, chances are that you will want to
7983 have some automatic simplifications or canonicalizations performed on them.
7984 This is done in the evaluation member function @code{eval()}. Let's say that
7985 we wanted all strings automatically converted to lowercase with
7986 non-alphabetic characters stripped, and empty strings removed:
7989 class mystring : public basic
7993 ex eval(int level = 0) const;
7997 ex mystring::eval(int level) const
8000 for (int i=0; i<str.length(); i++) @{
8002 if (c >= 'A' && c <= 'Z')
8003 new_str += tolower(c);
8004 else if (c >= 'a' && c <= 'z')
8008 if (new_str.length() == 0)
8011 return mystring(new_str).hold();
8015 The @code{level} argument is used to limit the recursion depth of the
8016 evaluation. We don't have any subexpressions in the @code{mystring}
8017 class so we are not concerned with this. If we had, we would call the
8018 @code{eval()} functions of the subexpressions with @code{level - 1} as
8019 the argument if @code{level != 1}. The @code{hold()} member function
8020 sets a flag in the object that prevents further evaluation. Otherwise
8021 we might end up in an endless loop. When you want to return the object
8022 unmodified, use @code{return this->hold();}.
8024 Let's confirm that it works:
8027 ex e = mystring("Hello, world!") + mystring("!?#");
8031 e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
8036 @subsection Optional member functions
8038 We have implemented only a small set of member functions to make the class
8039 work in the GiNaC framework. There are two functions that are not strictly
8040 required but will make operations with objects of the class more efficient:
8042 @cindex @code{calchash()}
8043 @cindex @code{is_equal_same_type()}
8045 unsigned calchash() const;
8046 bool is_equal_same_type(const basic &other) const;
8049 The @code{calchash()} method returns an @code{unsigned} hash value for the
8050 object which will allow GiNaC to compare and canonicalize expressions much
8051 more efficiently. You should consult the implementation of some of the built-in
8052 GiNaC classes for examples of hash functions. The default implementation of
8053 @code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
8054 class and all subexpressions that are accessible via @code{op()}.
8056 @code{is_equal_same_type()} works like @code{compare_same_type()} but only
8057 tests for equality without establishing an ordering relation, which is often
8058 faster. The default implementation of @code{is_equal_same_type()} just calls
8059 @code{compare_same_type()} and tests its result for zero.
8061 @subsection Other member functions
8063 For a real algebraic class, there are probably some more functions that you
8064 might want to provide:
8067 bool info(unsigned inf) const;
8068 ex evalf(int level = 0) const;
8069 ex series(const relational & r, int order, unsigned options = 0) const;
8070 ex derivative(const symbol & s) const;
8073 If your class stores sub-expressions (see the scalar product example in the
8074 previous section) you will probably want to override
8076 @cindex @code{let_op()}
8079 ex op(size_t i) const;
8080 ex & let_op(size_t i);
8081 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
8082 ex map(map_function & f) const;
8085 @code{let_op()} is a variant of @code{op()} that allows write access. The
8086 default implementations of @code{subs()} and @code{map()} use it, so you have
8087 to implement either @code{let_op()}, or @code{subs()} and @code{map()}.
8089 You can, of course, also add your own new member functions. Remember
8090 that the RTTI may be used to get information about what kinds of objects
8091 you are dealing with (the position in the class hierarchy) and that you
8092 can always extract the bare object from an @code{ex} by stripping the
8093 @code{ex} off using the @code{ex_to<mystring>(e)} function when that
8094 should become a need.
8096 That's it. May the source be with you!
8099 @node A comparison with other CAS, Advantages, Adding classes, Top
8100 @c node-name, next, previous, up
8101 @chapter A Comparison With Other CAS
8104 This chapter will give you some information on how GiNaC compares to
8105 other, traditional Computer Algebra Systems, like @emph{Maple},
8106 @emph{Mathematica} or @emph{Reduce}, where it has advantages and
8107 disadvantages over these systems.
8110 * Advantages:: Strengths of the GiNaC approach.
8111 * Disadvantages:: Weaknesses of the GiNaC approach.
8112 * Why C++?:: Attractiveness of C++.
8115 @node Advantages, Disadvantages, A comparison with other CAS, A comparison with other CAS
8116 @c node-name, next, previous, up
8119 GiNaC has several advantages over traditional Computer
8120 Algebra Systems, like
8125 familiar language: all common CAS implement their own proprietary
8126 grammar which you have to learn first (and maybe learn again when your
8127 vendor decides to `enhance' it). With GiNaC you can write your program
8128 in common C++, which is standardized.
8132 structured data types: you can build up structured data types using
8133 @code{struct}s or @code{class}es together with STL features instead of
8134 using unnamed lists of lists of lists.
8137 strongly typed: in CAS, you usually have only one kind of variables
8138 which can hold contents of an arbitrary type. This 4GL like feature is
8139 nice for novice programmers, but dangerous.
8142 development tools: powerful development tools exist for C++, like fancy
8143 editors (e.g. with automatic indentation and syntax highlighting),
8144 debuggers, visualization tools, documentation generators@dots{}
8147 modularization: C++ programs can easily be split into modules by
8148 separating interface and implementation.
8151 price: GiNaC is distributed under the GNU Public License which means
8152 that it is free and available with source code. And there are excellent
8153 C++-compilers for free, too.
8156 extendable: you can add your own classes to GiNaC, thus extending it on
8157 a very low level. Compare this to a traditional CAS that you can
8158 usually only extend on a high level by writing in the language defined
8159 by the parser. In particular, it turns out to be almost impossible to
8160 fix bugs in a traditional system.
8163 multiple interfaces: Though real GiNaC programs have to be written in
8164 some editor, then be compiled, linked and executed, there are more ways
8165 to work with the GiNaC engine. Many people want to play with
8166 expressions interactively, as in traditional CASs. Currently, two such
8167 windows into GiNaC have been implemented and many more are possible: the
8168 tiny @command{ginsh} that is part of the distribution exposes GiNaC's
8169 types to a command line and second, as a more consistent approach, an
8170 interactive interface to the Cint C++ interpreter has been put together
8171 (called GiNaC-cint) that allows an interactive scripting interface
8172 consistent with the C++ language. It is available from the usual GiNaC
8176 seamless integration: it is somewhere between difficult and impossible
8177 to call CAS functions from within a program written in C++ or any other
8178 programming language and vice versa. With GiNaC, your symbolic routines
8179 are part of your program. You can easily call third party libraries,
8180 e.g. for numerical evaluation or graphical interaction. All other
8181 approaches are much more cumbersome: they range from simply ignoring the
8182 problem (i.e. @emph{Maple}) to providing a method for `embedding' the
8183 system (i.e. @emph{Yacas}).
8186 efficiency: often large parts of a program do not need symbolic
8187 calculations at all. Why use large integers for loop variables or
8188 arbitrary precision arithmetics where @code{int} and @code{double} are
8189 sufficient? For pure symbolic applications, GiNaC is comparable in
8190 speed with other CAS.
8195 @node Disadvantages, Why C++?, Advantages, A comparison with other CAS
8196 @c node-name, next, previous, up
8197 @section Disadvantages
8199 Of course it also has some disadvantages:
8204 advanced features: GiNaC cannot compete with a program like
8205 @emph{Reduce} which exists for more than 30 years now or @emph{Maple}
8206 which grows since 1981 by the work of dozens of programmers, with
8207 respect to mathematical features. Integration, factorization,
8208 non-trivial simplifications, limits etc. are missing in GiNaC (and are
8209 not planned for the near future).
8212 portability: While the GiNaC library itself is designed to avoid any
8213 platform dependent features (it should compile on any ANSI compliant C++
8214 compiler), the currently used version of the CLN library (fast large
8215 integer and arbitrary precision arithmetics) can only by compiled
8216 without hassle on systems with the C++ compiler from the GNU Compiler
8217 Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
8218 macros to let the compiler gather all static initializations, which
8219 works for GNU C++ only. Feel free to contact the authors in case you
8220 really believe that you need to use a different compiler. We have
8221 occasionally used other compilers and may be able to give you advice.}
8222 GiNaC uses recent language features like explicit constructors, mutable
8223 members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
8224 literally. Recent GCC versions starting at 2.95.3, although itself not
8225 yet ANSI compliant, support all needed features.
8230 @node Why C++?, Internal structures, Disadvantages, A comparison with other CAS
8231 @c node-name, next, previous, up
8234 Why did we choose to implement GiNaC in C++ instead of Java or any other
8235 language? C++ is not perfect: type checking is not strict (casting is
8236 possible), separation between interface and implementation is not
8237 complete, object oriented design is not enforced. The main reason is
8238 the often scolded feature of operator overloading in C++. While it may
8239 be true that operating on classes with a @code{+} operator is rarely
8240 meaningful, it is perfectly suited for algebraic expressions. Writing
8241 @math{3x+5y} as @code{3*x+5*y} instead of
8242 @code{x.times(3).plus(y.times(5))} looks much more natural.
8243 Furthermore, the main developers are more familiar with C++ than with
8244 any other programming language.
8247 @node Internal structures, Expressions are reference counted, Why C++? , Top
8248 @c node-name, next, previous, up
8249 @appendix Internal structures
8252 * Expressions are reference counted::
8253 * Internal representation of products and sums::
8256 @node Expressions are reference counted, Internal representation of products and sums, Internal structures, Internal structures
8257 @c node-name, next, previous, up
8258 @appendixsection Expressions are reference counted
8260 @cindex reference counting
8261 @cindex copy-on-write
8262 @cindex garbage collection
8263 In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
8264 where the counter belongs to the algebraic objects derived from class
8265 @code{basic} but is maintained by the smart pointer class @code{ptr}, of
8266 which @code{ex} contains an instance. If you understood that, you can safely
8267 skip the rest of this passage.
8269 Expressions are extremely light-weight since internally they work like
8270 handles to the actual representation. They really hold nothing more
8271 than a pointer to some other object. What this means in practice is
8272 that whenever you create two @code{ex} and set the second equal to the
8273 first no copying process is involved. Instead, the copying takes place
8274 as soon as you try to change the second. Consider the simple sequence
8279 #include <ginac/ginac.h>
8280 using namespace std;
8281 using namespace GiNaC;
8285 symbol x("x"), y("y"), z("z");
8288 e1 = sin(x + 2*y) + 3*z + 41;
8289 e2 = e1; // e2 points to same object as e1
8290 cout << e2 << endl; // prints sin(x+2*y)+3*z+41
8291 e2 += 1; // e2 is copied into a new object
8292 cout << e2 << endl; // prints sin(x+2*y)+3*z+42
8296 The line @code{e2 = e1;} creates a second expression pointing to the
8297 object held already by @code{e1}. The time involved for this operation
8298 is therefore constant, no matter how large @code{e1} was. Actual
8299 copying, however, must take place in the line @code{e2 += 1;} because
8300 @code{e1} and @code{e2} are not handles for the same object any more.
8301 This concept is called @dfn{copy-on-write semantics}. It increases
8302 performance considerably whenever one object occurs multiple times and
8303 represents a simple garbage collection scheme because when an @code{ex}
8304 runs out of scope its destructor checks whether other expressions handle
8305 the object it points to too and deletes the object from memory if that
8306 turns out not to be the case. A slightly less trivial example of
8307 differentiation using the chain-rule should make clear how powerful this
8312 symbol x("x"), y("y");
8316 ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
8317 cout << e1 << endl // prints x+3*y
8318 << e2 << endl // prints (x+3*y)^3
8319 << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
8323 Here, @code{e1} will actually be referenced three times while @code{e2}
8324 will be referenced two times. When the power of an expression is built,
8325 that expression needs not be copied. Likewise, since the derivative of
8326 a power of an expression can be easily expressed in terms of that
8327 expression, no copying of @code{e1} is involved when @code{e3} is
8328 constructed. So, when @code{e3} is constructed it will print as
8329 @code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
8330 holds a reference to @code{e2} and the factor in front is just
8333 As a user of GiNaC, you cannot see this mechanism of copy-on-write
8334 semantics. When you insert an expression into a second expression, the
8335 result behaves exactly as if the contents of the first expression were
8336 inserted. But it may be useful to remember that this is not what
8337 happens. Knowing this will enable you to write much more efficient
8338 code. If you still have an uncertain feeling with copy-on-write
8339 semantics, we recommend you have a look at the
8340 @uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
8341 Marshall Cline. Chapter 16 covers this issue and presents an
8342 implementation which is pretty close to the one in GiNaC.
8345 @node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
8346 @c node-name, next, previous, up
8347 @appendixsection Internal representation of products and sums
8349 @cindex representation
8352 @cindex @code{power}
8353 Although it should be completely transparent for the user of
8354 GiNaC a short discussion of this topic helps to understand the sources
8355 and also explain performance to a large degree. Consider the
8356 unexpanded symbolic expression
8358 $2d^3 \left( 4a + 5b - 3 \right)$
8361 @math{2*d^3*(4*a+5*b-3)}
8363 which could naively be represented by a tree of linear containers for
8364 addition and multiplication, one container for exponentiation with base
8365 and exponent and some atomic leaves of symbols and numbers in this
8370 @cindex pair-wise representation
8371 However, doing so results in a rather deeply nested tree which will
8372 quickly become inefficient to manipulate. We can improve on this by
8373 representing the sum as a sequence of terms, each one being a pair of a
8374 purely numeric multiplicative coefficient and its rest. In the same
8375 spirit we can store the multiplication as a sequence of terms, each
8376 having a numeric exponent and a possibly complicated base, the tree
8377 becomes much more flat:
8381 The number @code{3} above the symbol @code{d} shows that @code{mul}
8382 objects are treated similarly where the coefficients are interpreted as
8383 @emph{exponents} now. Addition of sums of terms or multiplication of
8384 products with numerical exponents can be coded to be very efficient with
8385 such a pair-wise representation. Internally, this handling is performed
8386 by most CAS in this way. It typically speeds up manipulations by an
8387 order of magnitude. The overall multiplicative factor @code{2} and the
8388 additive term @code{-3} look somewhat out of place in this
8389 representation, however, since they are still carrying a trivial
8390 exponent and multiplicative factor @code{1} respectively. Within GiNaC,
8391 this is avoided by adding a field that carries an overall numeric
8392 coefficient. This results in the realistic picture of internal
8395 $2d^3 \left( 4a + 5b - 3 \right)$:
8398 @math{2*d^3*(4*a+5*b-3)}:
8404 This also allows for a better handling of numeric radicals, since
8405 @code{sqrt(2)} can now be carried along calculations. Now it should be
8406 clear, why both classes @code{add} and @code{mul} are derived from the
8407 same abstract class: the data representation is the same, only the
8408 semantics differs. In the class hierarchy, methods for polynomial
8409 expansion and the like are reimplemented for @code{add} and @code{mul},
8410 but the data structure is inherited from @code{expairseq}.
8413 @node Package tools, ginac-config, Internal representation of products and sums, Top
8414 @c node-name, next, previous, up
8415 @appendix Package tools
8417 If you are creating a software package that uses the GiNaC library,
8418 setting the correct command line options for the compiler and linker
8419 can be difficult. GiNaC includes two tools to make this process easier.
8422 * ginac-config:: A shell script to detect compiler and linker flags.
8423 * AM_PATH_GINAC:: Macro for GNU automake.
8427 @node ginac-config, AM_PATH_GINAC, Package tools, Package tools
8428 @c node-name, next, previous, up
8429 @section @command{ginac-config}
8430 @cindex ginac-config
8432 @command{ginac-config} is a shell script that you can use to determine
8433 the compiler and linker command line options required to compile and
8434 link a program with the GiNaC library.
8436 @command{ginac-config} takes the following flags:
8440 Prints out the version of GiNaC installed.
8442 Prints '-I' flags pointing to the installed header files.
8444 Prints out the linker flags necessary to link a program against GiNaC.
8445 @item --prefix[=@var{PREFIX}]
8446 If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
8447 (And of exec-prefix, unless @code{--exec-prefix} is also specified)
8448 Otherwise, prints out the configured value of @env{$prefix}.
8449 @item --exec-prefix[=@var{PREFIX}]
8450 If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
8451 Otherwise, prints out the configured value of @env{$exec_prefix}.
8454 Typically, @command{ginac-config} will be used within a configure
8455 script, as described below. It, however, can also be used directly from
8456 the command line using backquotes to compile a simple program. For
8460 c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
8463 This command line might expand to (for example):
8466 cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
8467 -lginac -lcln -lstdc++
8470 Not only is the form using @command{ginac-config} easier to type, it will
8471 work on any system, no matter how GiNaC was configured.
8474 @node AM_PATH_GINAC, Configure script options, ginac-config, Package tools
8475 @c node-name, next, previous, up
8476 @section @samp{AM_PATH_GINAC}
8477 @cindex AM_PATH_GINAC
8479 For packages configured using GNU automake, GiNaC also provides
8480 a macro to automate the process of checking for GiNaC.
8483 AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
8484 [, @var{ACTION-IF-NOT-FOUND}]]])
8492 Determines the location of GiNaC using @command{ginac-config}, which is
8493 either found in the user's path, or from the environment variable
8494 @env{GINACLIB_CONFIG}.
8497 Tests the installed libraries to make sure that their version
8498 is later than @var{MINIMUM-VERSION}. (A default version will be used
8502 If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
8503 to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
8504 variable to the output of @command{ginac-config --libs}, and calls
8505 @samp{AC_SUBST()} for these variables so they can be used in generated
8506 makefiles, and then executes @var{ACTION-IF-FOUND}.
8509 If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
8510 @env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
8514 This macro is in file @file{ginac.m4} which is installed in
8515 @file{$datadir/aclocal}. Note that if automake was installed with a
8516 different @samp{--prefix} than GiNaC, you will either have to manually
8517 move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
8518 aclocal the @samp{-I} option when running it.
8521 * Configure script options:: Configuring a package that uses AM_PATH_GINAC.
8522 * Example package:: Example of a package using AM_PATH_GINAC.
8526 @node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
8527 @c node-name, next, previous, up
8528 @subsection Configuring a package that uses @samp{AM_PATH_GINAC}
8530 Simply make sure that @command{ginac-config} is in your path, and run
8531 the configure script.
8538 The directory where the GiNaC libraries are installed needs
8539 to be found by your system's dynamic linker.
8541 This is generally done by
8544 editing @file{/etc/ld.so.conf} and running @command{ldconfig}
8550 setting the environment variable @env{LD_LIBRARY_PATH},
8553 or, as a last resort,
8556 giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
8557 running configure, for instance:
8560 LDFLAGS=-R/home/cbauer/lib ./configure
8565 You can also specify a @command{ginac-config} not in your path by
8566 setting the @env{GINACLIB_CONFIG} environment variable to the
8567 name of the executable
8570 If you move the GiNaC package from its installed location,
8571 you will either need to modify @command{ginac-config} script
8572 manually to point to the new location or rebuild GiNaC.
8583 --with-ginac-prefix=@var{PREFIX}
8584 --with-ginac-exec-prefix=@var{PREFIX}
8587 are provided to override the prefix and exec-prefix that were stored
8588 in the @command{ginac-config} shell script by GiNaC's configure. You are
8589 generally better off configuring GiNaC with the right path to begin with.
8593 @node Example package, Bibliography, Configure script options, AM_PATH_GINAC
8594 @c node-name, next, previous, up
8595 @subsection Example of a package using @samp{AM_PATH_GINAC}
8597 The following shows how to build a simple package using automake
8598 and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
8602 #include <ginac/ginac.h>
8606 GiNaC::symbol x("x");
8607 GiNaC::ex a = GiNaC::sin(x);
8608 std::cout << "Derivative of " << a
8609 << " is " << a.diff(x) << std::endl;
8614 You should first read the introductory portions of the automake
8615 Manual, if you are not already familiar with it.
8617 Two files are needed, @file{configure.in}, which is used to build the
8621 dnl Process this file with autoconf to produce a configure script.
8623 AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
8629 AM_PATH_GINAC(0.9.0, [
8630 LIBS="$LIBS $GINACLIB_LIBS"
8631 CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
8632 ], AC_MSG_ERROR([need to have GiNaC installed]))
8637 The only command in this which is not standard for automake
8638 is the @samp{AM_PATH_GINAC} macro.
8640 That command does the following: If a GiNaC version greater or equal
8641 than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
8642 and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
8643 the error message `need to have GiNaC installed'
8645 And the @file{Makefile.am}, which will be used to build the Makefile.
8648 ## Process this file with automake to produce Makefile.in
8649 bin_PROGRAMS = simple
8650 simple_SOURCES = simple.cpp
8653 This @file{Makefile.am}, says that we are building a single executable,
8654 from a single source file @file{simple.cpp}. Since every program
8655 we are building uses GiNaC we simply added the GiNaC options
8656 to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
8657 want to specify them on a per-program basis: for instance by
8661 simple_LDADD = $(GINACLIB_LIBS)
8662 INCLUDES = $(GINACLIB_CPPFLAGS)
8665 to the @file{Makefile.am}.
8667 To try this example out, create a new directory and add the three
8670 Now execute the following commands:
8673 $ automake --add-missing
8678 You now have a package that can be built in the normal fashion
8687 @node Bibliography, Concept index, Example package, Top
8688 @c node-name, next, previous, up
8689 @appendix Bibliography
8694 @cite{ISO/IEC 14882:1998: Programming Languages: C++}
8697 @cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
8700 @cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley
8703 @cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley
8706 @cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
8707 and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts
8710 @cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
8711 James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
8712 Academic Press, London
8715 @cite{Computer Algebra Systems - A Practical Guide},
8716 Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
8719 @cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
8720 Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
8723 @cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
8724 ISBN 3-540-66572-2, 2001, Springer, Heidelberg
8727 @cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
8732 @node Concept index, , Bibliography, Top
8733 @c node-name, next, previous, up
8734 @unnumbered Concept index