Provided by: libmath-symbolic-perl_0.612-3_all bug

NAME

       Math::Symbolic - Symbolic calculations

SYNOPSIS

         use Math::Symbolic;

         my $tree = Math::Symbolic->parse_from_string('1/2 * m * v^2');
         # Now do symbolic calculations with $tree.
         # ... like deriving it...

         my ($sub) = Math::Symbolic::Compiler->compile_to_sub($tree);

         my $kinetic_energy = $sub->($mass, $velocity);

DESCRIPTION

       Math::Symbolic is intended to offer symbolic calculation capabilities to the Perl
       programmer without using external (and commercial) libraries and/or applications.

       Unless, however, some interested and knowledgable developers turn up to participate in the
       development, the library will be severely limited by my experience in the area. Symbolic
       calculations are an active field of research in CS.

       There are several ways to construct Math::Symbolic trees. There are no actual
       Math::Symbolic objects, but rather trees of objects of subclasses of Math::Symbolic. The
       most general but unfortunately also the least intuitive way of constructing trees is to
       use the constructors of the Math::Symbolic::Operator, Math::Symbolic::Variable, and
       Math::Symbolic::Constant classes to create (nested) objects of the corresponding types.

       Furthermore, you may use the overloaded interface to apply the standard Perl operators
       (and functions, see "OVERLOADED OPERATORS") to existing Math::Symbolic trees and standard
       Perl expressions.

       Possibly the most convenient way of constructing Math::Symbolic trees is using the builtin
       parser to generate trees from expressions such as "2 * x^5".  You may use the
       "Math::Symbolic->parse_from_string()" class method for this.

       Of course, you may combine the overloaded interface with the parser to generate trees with
       Perl code such as "$term * 5 * 'sin(omega*t+phi)'" which will create a tree of the
       existing tree $term times 5 times the sine of the vars omega times t plus phi.

       There are several modules in the distribution that contain subroutines related to
       calculus. These are not loaded by Math::Symbolic by default.  Furthermore, there are
       several extensions to Math::Symbolic available from CPAN as separate distributions. Please
       refer to "SEE ALSO" for an incomplete list of these.

       For example, Math::Symbolic::MiscCalculus come with "Math::Symbolic" and contains routines
       to compute Taylor Polynomials and the associated errors.

       Routines related to vector calculus such as grad, div, rot, and Jacobi- and Hesse matrices
       are available through the Math::Symbolic::VectorCalculus module. This module is also able
       to compute Taylor Polynomials of functions of two variables, directional derivatives,
       total differentials, and Wronskian Determinants.

       Some basic support for linear algebra can be found in Math::Symbolic::MiscAlgebra. This
       includes a routine to compute the determinant of a matrix of "Math::Symbolic" trees.

   EXPORT
       None by default, but you may choose to have the following constants exported to your
       namespace using the standard Exporter semantics.  There are two export tags: :all and
       :constants. :all will export all constants and the parse_from_string subroutine.

         Constants for transcendetal numbers:
           EULER (2.7182...)
           PI    (3.14159...)

         Constants representing operator types: (First letter indicates arity)
         (These evaluate to the same numbers that are returned by the type()
          method of Math::Symbolic::Operator objects.)
           B_SUM
           B_DIFFERENCE
           B_PRODUCT
           B_DIVISION
           B_LOG
           B_EXP
           U_MINUS
           U_P_DERIVATIVE (partial derivative)
           U_T_DERIVATIVE (total derivative)
           U_SINE
           U_COSINE
           U_TANGENT
           U_COTANGENT
           U_ARCSINE
           U_ARCCOSINE
           U_ARCTANGENT
           U_ARCCOTANGENT
           U_SINE_H
           U_COSINE_H
           U_AREASINE_H
           U_AREACOSINE_H
           B_ARCTANGENT_TWO

         Constants representing Math::Symbolic term types:
         (These evaluate to the same numbers that are returned by the term_type()
          methods.)
           T_OPERATOR
           T_CONSTANT
           T_VARIABLE

         Subroutines:
           parse_from_string (returns Math::Symbolic tree)

CLASS DATA

       The package variable $Parser will contain a Parse::RecDescent object that is used to parse
       strings at runtime.

SUBROUTINES

   parse_from_string
       This subroutine takes a string as argument and parses it using a Parse::RecDescent parser
       taken from the package variable $Math::Symbolic::Parser. It generates a Math::Symbolic
       tree from the string and returns that tree.

       The string may contain any identifiers matching /[a-zA-Z][a-zA-Z0-9_]*/ which will be
       parsed as variables of the corresponding name.

       Please refer to Math::Symbolic::Parser for more information.

EXAMPLES

       This example demonstrates variable and operator creation using object prototypes as well
       as partial derivatives and the various ways of applying derivatives and simplifying terms.
       Furthermore, it shows how to use the compiler for simple expressions.

         use Math::Symbolic qw/:all/;

         my $energy = parse_from_string(<<'HERE');
               kinetic(mass, velocity, time) +
               potential(mass, z, time)
         HERE

         $energy->implement(kinetic => '(1/2) * mass * velocity(time)^2');
         $energy->implement(potential => 'mass * g * z(t)');

         $energy->set_value(g => 9.81); # permanently

         print "Energy is: $energy\n";

         # Is how does the energy change with the height?
         my $derived = $energy->new('partial_derivative', $energy, 'z');
         $derived = $derived->apply_derivatives()->simplify();

         print "Changes with the heigth as: $derived\n";

         # With whatever values you fancy:
         print "Putting in some sample values: ",
               $energy->value(mass => 20, velocity => 10, z => 5),
               "\n";

         # Too slow?
         $energy->implement(g => '9.81'); # To get rid of the variable

         my ($sub) = Math::Symbolic::Compiler->compile($energy);

         print "This was much faster: ",
               $sub->(20, 10, 5),  # vars ordered alphabetically
               "\n";

OVERLOADED OPERATORS

       Since version 0.102, several arithmetic operators have been overloaded.

       That means you can do most arithmetic with Math::Symbolic trees just as if they were plain
       Perl scalars.

       The following operators are currently overloaded to produce valid Math::Symbolic trees
       when applied to an expression involving at least one Math::Symbolic object:

         +, -, *, /, **, sqrt, log, exp, sin, cos

       Furthermore, some contexts have been overloaded with particular behaviour: '""'
       (stringification context) has been overloaded to produce the string representation of the
       object. '0+' (numerical context) has been overloaded to produce the value of the object.
       'bool' (boolean context) has been overloaded to produce the value of the object.

       If one of the operands of an overloaded operator is a Math::Symbolic tree and the over is
       undef, the module will throw an error unless the operator is a + or a -. If the operator
       is an addition, the result will be the original Math::Symbolic tree. If the operator is a
       subtraction, the result will be the negative of the Math::Symbolic tree. Reason for this
       inconsistent behaviour is that it makes idioms like the following possible:

         @objects = (... list of Math::Symbolic trees ...);
         $sum += $_ foreach @objects;

       Without this behaviour, you would have to shift the first object into $sum before using
       it. This is not a problem in this case, but if you are applying some complex calculation
       to each object in the loop body before adding it to the sum, you'd have to either split
       the code into two loops or replicate the code required for the complex calculation when
       shift()ing the first object into $sum.

       Warning: The operator to use for exponentiation is the normal Perl operator for
       exponentiation "**", NOT the caret "^" which denotes exponentiation in the notation that
       is recognized by the Math::Symbolic parsers! The "^" operator will be interpreted as the
       normal binary xor.

EXTENDING THE MODULE

       Due to several design decisions, it is probably rather difficult to extend the
       Math::Symbolic related modules through subclassing. Instead, we chose to make the module
       extendable through delegation.

       That means you can introduce your own methods to extend Math::Symbolic's functionality.
       How this works in detail can be read in Math::Symbolic::Custom.

       Some of the extensions available via CPAN right now are listed in the "SEE ALSO" section.

PERFORMANCE

       Math::Symbolic can become quite slow if you use it wrong. To be honest, it can even be
       slow if you use it correctly. This section is meant to give you an idea about what you can
       do to have Math::Symbolic compute as quickly as possible. It has some explanation and a
       couple of 'red flags' to watch out for.  We'll focus on two central points: Creation and
       evaluation.

   CREATING Math::Symbolic TREES
       Math::Symbolic provides several means of generating Math::Symbolic trees (which are just
       trees of Math::Symbolic::Constant, Math::Symbolic::Variable and most importantly
       Math::Symbolic::Operator objects).

       The most convenient way is to use the builtin parser (for example via the
       "parse_from_string()" subroutine). Problem is, this darn thing becomes really slow for
       long input strings. This is a known problem for Parse::RecDescent parsers and the
       Math::Symbolic grammar isn't the shortest either.

       Try to break the formulas you parse into smallish bits. Test the parser performance to see
       how small they need to be.

       I'll give a simple example where this first advice is gospel:

         use Math::Symbolic qw/parse_from_string/;
         my @formulas;
         foreach my $var (qw/x y z foo bar baz/) {
             my $formula = parse_from_string("sin(x)*$var+3*y^z-$var*x");
             push @formulas, $formula;
         }

       So what's wrong here? I'm parsing the whole formula every time. How about this?

         use Math::Symbolic qw/parse_from_string/;
         my @formulas;
         my $sin = parse_from_string('sin(x)');
         my $term = parse_from_string('3*y^z');
         my $x = Math::Symbolic::Variable->new('x');
         foreach my $var (qw/x y z foo bar baz/) {
                 my $v = $x->new($var);
             my $formula = $sin*$var + $term - $var*$x;
             push @formulas, $formula;
         }

       I wouldn't call that more legible, but you notice how I moved all the heavy lifting out of
       the loop. You'll know and do this for normal code, but it's maybe not as obvious when
       dealing with such code. Now, since this is still slow and - if anything - ugly, we'll do
       something really clever now to get the best of both worlds!

         use Math::Symbolic qw/parse_from_string/;
         my @formulas;
         my $proto = parse_from_string('sin(x)*var+3*y^z-var*x");
         foreach my $var (qw/x y z foo bar baz/) {
             my $formula = $proto->new();
             $formula->implement(var => Math::Symbolic::Variable->new($var));
             push @formulas, $formula;
         }

       Notice how we can combine legibility of a clean formula with removing all parsing work
       from the loop? The "implement()" method is described in detail in Math::Symbolic::Base.

       On a side note: One thing you could do to bring your computer to its knees is to take a
       function like sin(a*x)*cos(b*x)/e^(2*x), derive that in respect to x a couple of times
       (like, erm, 50 times?), call "to_string()" on it and parse that string again.

       Almost as convenient as the parser is the overloaded interface.  That means, you create a
       Math::Symbolic object and use it in algebraic expressions as if it was a variable or
       number. This way, you can even multiply a Math::Symbolic tree with a string and have the
       string be parsed as a subtree.  Example:

         my $x = Math::Symbolic::Variable->new('x');
         my $formula = $x - sin(3*$x); # $formula will be a M::S tree
         # or:
         my $another = $x - 'sin(3*x)'; # have the string parsed as M::S tree

       This, however, turns out to be rather slow, too. It is only about two to five times faster
       than parsing the formula all the way.

       Use the overloaded interface to construct trees from existing Math::Symbolic objects, but
       if you need to create new trees quickly, resort to building them by hand.

       Finally, you can create objects using the "new()" constructors from
       Math::Symbolic::Operator and friends. These can be called in two forms, a long one that
       gives you complete control (signature for variables, etc.)  and a short hand. Even if it
       is just to protect your finger tips from burning, you should use the short hand whenever
       possible. It is also slightly faster.

       Use the constructors to build Math::Symbolic trees if you need speed.  Using a prototype
       object and calling "new()" on that may help with the typing effort and should not result
       in a slow down.

   CRUNCHING NUMBERS WITH Math::Symbolic
       As with the generation of Math::Symbolic trees, the evaluation of a formula can be done in
       distinct ways.

       The simplest is, of course, to call "value()" on the tree and have that calculate the
       value of the formula. You might have to supply some input values to the formula via
       "value()", but you can also call "set_value()" before using "value()". But that's not
       faster.  For each call to "value()", the computer walks the complete Math::Symbolic tree
       and evaluates the nodes. If it reaches a leaf, the resulting value is propagated back up
       the tree. (It's a depth-first search.)

       Calling value() on a Math::Symbolic tree requires walking the tree for every evaluation of
       the formula. Use this if you'll evaluate the formula only a few times.

       You may be able to make the formula simpler using the Math::Symbolic simplification
       routines (like "simplify()" or some stuff in the Math::Symbolic::Custom::* modules).
       Simpler formula are quicker to evaluate.  In particular, the simplification should fold
       constants.

       If you're going to evaluate a tree many times, try simplifying it first.

       But again, your mileage may vary. Test first.

       If the overhead of calling "value()" is unaccepable, you should use the
       Math::Symbolic::Compiler to compile the tree to Perl code. (Which usually comes in
       compiled form as an anonymous subroutine.) Example:

         my $tree = parse_from_string('3*x+sin(y)^(z+1)');
         my $sub = $tree->to_sub(y => 0, x => 1, z => 2);
         foreach (1..100) {
           # define $x, $y, and $z
           my $res = $sub->($y, $x, $z);
           # faster than $tree->value(x => $x, y => $y, z => $z) !!!
         }

       Compile your Math::Symbolic trees to Perl subroutines for evaluation in tight loops. The
       speedup is in the range of a few thousands.

       On an interesting side note, the subroutines compiled from Math::Symbolic trees are just
       as fast as hand-crafted, "performance tuned" subroutines.

       If you have extremely long formulas, you can choose to even resort to more extreme
       measures than generating Perl code. You can have Math::Symbolic generate C code for you,
       compile that and link it into your application at run time. It will then be available to
       you as a subroutine.

       This is not the most portable thing to do. (You need Inline::C which in turn needs the C
       compiler that was used to compile your perl.)  Therefore, you need to install an extra
       module for this. It's called Math::Symbolic::Custom::CCompiler. The speed-up for short
       formulas is only about factor 2 due to the overhead of calling the Perl subroutine, but
       with sufficiently complicated formulas, you should be able to get a boost up to factor 100
       or even 1000.

       For raw execution speed, compile your trees to C code using
       Math::Symbolic::Custom::CCompiler.

   PROOF
       In the last two sections, you were told a lot about the performance of two important
       aspects of Math::Symbolic handling. But eventhough benchmarks are very system dependent
       and have limited meaning to the general case, I'll supply some proof for what I claimed.
       This is Perl 5.8.6 on linux-2.6.9, x86_64 (Athlon64 3200+).

       In the following tables, value means evaluation using the "value()" method, eval means
       evaluation of Perl code as a string, sub is a hand-crafted Perl subroutine, compiled is
       the compiled Perl code, c is the compiled C code. Evaluation of a very simple function
       yields:

         f(x) = x*2
                       Rate    value     eval      sub compiled        c
         value      17322/s       --     -68%     -99%     -99%     -99%
         eval       54652/s     215%       --     -97%     -97%     -97%
         sub      1603578/s    9157%    2834%       --      -1%     -16%
         compiled 1616630/s    9233%    2858%       1%       --     -15%
         c        1907541/s   10912%    3390%      19%      18%       --

       We see that resorting to C is a waste in such simple cases. Compiling to a Perl sub,
       however is a good idea.

         f(x,y,z) = x*y*z+sin(x*y*z)-cos(x*y*z)
                       Rate    value     eval compiled      sub        c
         value       1993/s       --     -88%    -100%    -100%    -100%
         eval       16006/s     703%       --     -97%     -97%     -99%
         compiled  544217/s   27202%    3300%       --      -2%     -56%
         sub       556737/s   27830%    3378%       2%       --     -55%
         c        1232362/s   61724%    7599%     126%     121%       --

         f(x,y,z,a,b) = x^y^tan(a*z)^(y*sin(x^(z*b)))
                      Rate    value     eval compiled      sub        c
         value      2181/s       --     -84%     -99%     -99%    -100%
         eval      13613/s     524%       --     -97%     -97%     -98%
         compiled 394945/s   18012%    2801%       --      -5%     -48%
         sub      414328/s   18901%    2944%       5%       --     -46%
         c        763985/s   34936%    5512%      93%      84%       --

       These more involved examples show that using value() can become unpractical even if you're
       just doing a 2D plot with just a few thousand points.  The C routines aren't that much
       faster, but they scale much better.

       Now for something different. Let's see whether I lied about the creation of Math::Symbolic
       trees. parse indicates that the parser was used to create the object tree. long indicates
       that the long syntax of the constructor was used. short... well. proto means that the
       objects were created from prototypes of the same class. For ol_long and ol_parse, I used
       the overloaded interface in conjunction with constructors or parsing (a la "$x * 'y+z'").

         f(x) = x
                      Rate  parse  long   short  ol_long  ol_parse  proto
         parse       258/s     --  -100%  -100%    -100%     -100%  -100%
         long      95813/s 37102%     --   -33%     -34%      -34%   -35%
         short    143359/s 55563%    50%     --      -2%       -2%    -3%
         ol_long  146022/s 56596%    52%     2%       --       -0%    -1%
         ol_parse 146256/s 56687%    53%     2%       0%        --    -1%
         proto    147119/s 57023%    54%     3%       1%        1%     --

       Obviously, the parser gets blown to pieces, performance-wise. If you want to use it, but
       cannot accept its tranquility, you can resort to Math::SymbolicX::Inline and have the
       formulas parsed at compile time. (Which isn't faster, but means that they are available
       when the program runs.)  All other methods are about the same speed. Note, that the ol_*
       tests are just the same as short here, because in case of "f(x) = x", you cannot make use
       of the overloaded interface.

         f(x,y,a,b) = x*y(a,b)
                     Rate  parse  ol_parse ol_long   long  proto  short
         parse      125/s     --      -41%    -41%  -100%  -100%  -100%
         ol_parse   213/s    70%        --     -0%   -99%   -99%   -99%
         ol_long    213/s    70%        0%      --   -99%   -99%   -99%
         long     26180/s 20769%    12178%  12171%     --    -6%   -10%
         proto    27836/s 22089%    12955%  12947%     6%     --    -5%
         short    29148/s 23135%    13570%  13562%    11%     5%     --

         f(x,a) = sin(x+a)*3-5*x
                     Rate    parse ol_long ol_parse     proto     short
         parse     41.2/s       --    -83%     -84%     -100%     -100%
         ol_long    250/s     505%      --      -0%      -97%      -98%
         ol_parse   250/s     506%      0%       --      -97%      -98%
         proto     9779/s   23611%   3819%    3810%        --       -3%
         short    10060/s   24291%   3932%    3922%        3%        --

       The picture changes when we're dealing with slightly longer functions.  The performance of
       the overloaded interface isn't that much better than the parser. (Since it uses the parser
       to convert non-Math::Symbolic operands.)  ol_long should, however, be faster than
       ol_parse. I'll refine the benchmark somewhen. The three other construction methods are
       still about the same speed. I omitted the long version in the last benchmark because the
       typing work involved was unnerving.

SEE ALSO

       New versions of this module can be found on http://steffen-mueller.net or CPAN. The module
       development takes place on Sourceforge at http://sourceforge.net/projects/math-symbolic/

       The following modules come with this distribution:

       Math::Symbolic::ExportConstants, Math::Symbolic::AuxFunctions

       Math::Symbolic::Base, Math::Symbolic::Operator, Math::Symbolic::Constant,
       Math::Symbolic::Variable

       Math::Symbolic::Custom, Math::Symbolic::Custom::Base,
       Math::Symbolic::Custom::DefaultTests, Math::Symbolic::Custom::DefaultMods
       Math::Symbolic::Custom::DefaultDumpers

       Math::Symbolic::Derivative, Math::Symbolic::MiscCalculus, Math::Symbolic::VectorCalculus,
       Math::Symbolic::MiscAlgebra

       Math::Symbolic::Parser, Math::Symbolic::Parser::Precompiled, Math::Symbolic::Compiler

       The following modules are extensions on CPAN that do not come with this distribution in
       order to keep the distribution size reasonable.

       Math::SymbolicX::Inline - (Inlined Math::Symbolic functions)

       Math::Symbolic::Custom::CCompiler (Compile Math::Symbolic trees to C for speed or for use
       in C code)

       Math::SymbolicX::BigNum (Big number support for the Math::Symbolic parser)

       Math::SymbolicX::Complex (Complex number support for the Math::Symbolic parser)

       Math::Symbolic::Custom::Contains (Find subtrees in Math::Symbolic expressions)

       Math::SymbolicX::ParserExtensionFactory (Generate parser extensions for the Math::Symbolic
       parser)

       Math::Symbolic::Custom::ErrorPropagation (Calculate Gaussian Error Propagation)

       Math::SymbolicX::Statistics::Distributions (Statistical Distributions as Math::Symbolic
       functions)

       Math::SymbolicX::NoSimplification (Turns off Math::Symbolic simplifications)

AUTHOR

       Please send feedback, bug reports, and support requests to the Math::Symbolic support
       mailing list: math-symbolic-support at lists dot sourceforge dot net. Please consider
       letting us know how you use Math::Symbolic. Thank you.

       If you're interested in helping with the development or extending the module's
       functionality, please contact the developers' mailing list: math-symbolic-develop at lists
       dot sourceforge dot net.

       List of contributors:

         Steffen MXller, smueller at cpan dot org
         Stray Toaster, mwk at users dot sourceforge dot net
         Oliver EbenhXh

COPYRIGHT AND LICENSE

       Copyright (C) 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2013 by Steffen
       Mueller

       This library is free software; you can redistribute it and/or modify it under the same
       terms as Perl itself, either Perl version 5.6.1 or, at your option, any later version of
       Perl 5 you may have available.