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 availlable 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 availlable 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
  
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.

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 availlable 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 availlable 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 ressort to Math::SymbolicX::Inline and have the formulas parsed at compile time. (Which isn't faster, but means that they are availlable 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.

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 Müller, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net
Oliver Ebenhöh

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