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Math::Symbolic

Module Version: 0.612
Math::Symbolic - Symbolic calculations

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);

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.

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)

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

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.

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";

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.

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.

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.

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**.

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.**

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.

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)

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 M�ller, smueller at cpan dot org Stray Toaster, mwk at users dot sourceforge dot net Oliver Ebenh�h

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.

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