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

Provided by: libmath-symbolic-perl_0.612-3_all

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

NAME

SYNOPSIS

DESCRIPTION

SUBROUTINES

AUTHOR

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