Manual Reference Pages - MATH::SYMBOLIC::MISCCALCULUS (3)
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;
$taylor_poly = TaylorPolynomial $function, $degree, $variable, $pos;
$lagrange_error = TaylorErrorLagrange $function, $degree, $variable;
$lagrange_error = TaylorErrorLagrange $function, $degree, $variable, $pos;
$lagrange_error = TaylorErrorLagrange $function, $degree, $variable, $pos,
# 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:
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
Tn(f, x, x0) =
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. Thats 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.
Its 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
Polynomials 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 isnt specified explicitly, the
variable will be named theta as in the formula above.
TaylorErrorCauchy computes and returns the formula for the Taylor
Polynomials 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
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 youre interested in helping with the development or extending the
modules 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
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/
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