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 MXller, symbolic-module at steffen-mueller dot net
         Stray Toaster, mwk at users dot sourceforge dot net
         Oliver EbenhXh

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