Math::Symbolic::MiscCalculus - Miscellaneous calculus routines (eg Taylor poly)
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;
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.
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
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 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 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.
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, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net
Oliver Ebenh�h
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