Math::Symbolic::MiscCaUseruContributed Perl DocMath::Symbolic::MiscCalculus(3)NAMEMath::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 particularily 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 MA~Xller, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net
Oliver EbenhA~Xh
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 v5.14.1 2011-07-26 Math::Symbolic::MiscCalculus(3)
