Math::Symbolic::MiscCalculus(3pm)			User Contributed Perl Documentation			 Math::Symbolic::MiscCalculus(3pm)

NAME

       Math::Symbolic::MiscCalculus - Miscellaneous calculus routines (eg Taylor poly)

SYNOPSIS

	 use Math::Symbolic qw/:all/;
	 use Math::Symbolic::MiscCalculus qw/:all/; # not loaded by Math::Symbolic

	 $taylor_poly = TaylorPolynomial $function, $degree, $variable;
	 # or:
	 $taylor_poly = TaylorPolynomial $function, $degree, $variable, $pos;

	 $lagrange_error = TaylorErrorLagrange $function, $degree, $variable;
	 # or:
	 $lagrange_error = TaylorErrorLagrange $function, $degree, $variable, $pos;
	 # or:
	 $lagrange_error = TaylorErrorLagrange $function, $degree, $variable, $pos,
					       $name_for_range_variable;

	 # This has the same syntax variations as the Lagrange error:
	 $cauchy_error = TaylorErrorLagrange $function, $degree, $variable;

DESCRIPTION

       This module provides several subroutines related to calculus such as computing Taylor polynomials and errors the associated errors from
       Math::Symbolic trees.

       Please note that the code herein may or may not be refactored into the OO-interface of the Math::Symbolic module in the future.

   EXPORT
       None by default.

       You may choose to have any of the following routines exported to the calling namespace. ':all' tag exports all of the following:

	 TaylorPolynomial
	 TaylorErrorLagrange
	 TaylorErrorCauchy

SUBROUTINES

   TaylorPolynomial
       This function (symbolically) computes the nth-degree Taylor Polynomial of a given function. Generally speaking, the Taylor Polynomial is an
       n-th degree polynomial that approximates the original function. It does so particularly well in the proximity of a certain point x0.
       (Since my mathematical English jargon is lacking, I strongly suggest you read up on what this is in a book.)

       Mathematically speaking, the Taylor Polynomial of the function f(x) looks like this:

	 Tn(f, x, x0) =
	   sum_from_k=0_to_n(
	       n-th_total_derivative(f)(x0) / k! * (x-x0)^k
	   )

       First argument to the subroutine must be the function to approximate. It may be given either as a string to be parsed or as a valid
       Math::Symbolic tree.  Second argument must be an integer indicating to which degree to approximate.  The third argument is the last
       required argument and denotes the variable to use for approximation either as a string (name) or as a Math::Symbolic::Variable object.
       That's the 'x' above.  The fourth argument is optional and specifies the name of the variable to introduce as the point of approximation.
       May also be a variable object.  It's the 'x0' above. If not specified, the name of this variable will be assumed to be the name of the
       function variable (the 'x') with '_0' appended.

       This routine is for functions of one variable only. There is an equivalent for functions of two variables in the
       Math::Symbolic::VectorCalculus package.

   TaylorErrorLagrange
       TaylorErrorLagrange computes and returns the formula for the Taylor Polynomial's approximation error after Lagrange. (Again, my English
       terminology is lacking.) It looks similar to this:

	 Rn(f, x, x0) =
	   n+1-th_total_derivative(f)( x0 + theta * (x-x0) ) / (n+1)! * (x-x0)^(n+1)

       Please refer to your favourite book on the topic. 'theta' may be any number between 0 and 1.

       The calling conventions for TaylorErrorLagrange are similar to those of TaylorPolynomial, but TaylorErrorLagrange takes an extra optional
       argument specifying the name of 'theta'. If it isn't specified explicitly, the variable will be named 'theta' as in the formula above.

   TaylorErrorCauchy
       TaylorErrorCauchy computes and returns the formula for the Taylor Polynomial's approximation error after (guess who!) Cauchy.  (Again, my
       English terminology is lacking.) It looks similar to this:

	 Rn(f, x, x0) = TaylorErrorLagrange(...) * (1 - theta)^n

       Please refer to your favourite book on the topic and the documentation for TaylorErrorLagrange. 'theta' may be any number between 0 and 1.

       The calling conventions for TaylorErrorCauchy are identical to those of TaylorErrorLagrange.

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 Mueller, symbolic-module at steffen-mueller dot net
	 Stray Toaster, mwk at users dot sourceforge dot net
	 Oliver Ebenhoeh

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/

       Math::Symbolic

perl v5.10.1							    2011-01-01					 Math::Symbolic::MiscCalculus(3pm)