Provided by: libmath-symbolic-perl_0.612-2_all 
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.