Discrete derivative polynomial function evaluation. More...

Functions/Subroutines
pure double precision function	evaluate_dd_polynomial (x, derivatives)
	Evaluate a polynomial at x, defined by its derivatives (starting from the \(0^th\) derivative)

pure double precision function	evaluate_dd_polynomial_derivative (x, derivatives, n)
	Evaluate the polynomial's \(N^{th}\) derivative (.

pure double precision function, dimension(size(derivatives))	get_dd_polynomial (derivatives)
	Get the polynomial associated to a DD.

pure double precision function, dimension(n)	get_dd_lagrange_vector (x, n)
	Get the vector made of the Xvector * ifactorials.

Variables
integer, parameter	max_factorials = 8

double precision, dimension(max_factorials), parameter	factorials = [ 1d0, 1d0, 2d0, 6d0, 24d0, 120d0, 720d0, 5040d0 ]

double precision, dimension(max_factorials), parameter	ifactorials = 1d0/factorials

Detailed Description

Discrete derivative polynomial function evaluation.

This DD polynomial module furnishes very useful function to evaluate the DD and its derivatives at a particular place x. Once one has computed the numerical value of the derivative at a particular point the evaluation is a simple computation of the Taylor expansion at that point.

For that purpose, we use the common evaluate_polynomial function by considering the \(i^th\) polynomial coefficient as \(p_i = \frac{D_i}{i!} \).

Function/Subroutine Documentation

◆ evaluate_dd_polynomial()

pure double precision function mod_dd_polynomial::evaluate_dd_polynomial	(	double precision, intent(in)	x,
		double precision, dimension(:), intent(in)	derivatives )

Evaluate a polynomial at x, defined by its derivatives (starting from the \(0^th\) derivative)

Parameters

[in]	x	the position where to evaluate
[in]	derivatives	an array such that \(derivatives(i)=\frac{d^{i-1}P}{dx^{i-1}}\)

◆ evaluate_dd_polynomial_derivative()

pure double precision function mod_dd_polynomial::evaluate_dd_polynomial_derivative	(	double precision, intent(in)	x,
		double precision, dimension(:), intent(in)	derivatives,
		integer, intent(in)	n )

Evaluate the polynomial's \(N^{th}\) derivative (.

See also: evaluate_dd_polynomial)

◆ get_dd_lagrange_vector()

pure double precision function, dimension(n) mod_dd_polynomial::get_dd_lagrange_vector	(	double precision, intent(in)	x,
		integer, intent(in)	n )

Get the vector made of the Xvector * ifactorials.

The Lagrange vector can be used when evaluating a discrete derivative or a simple Taylor expansion formulae. It is defined as:

\[ Lv_i = \frac{ x^{i-1} }{i!} \]

Parameters

[in]	x	the position
[in]	N	the size of the array @

◆ get_dd_polynomial()

pure double precision function, dimension(size(derivatives)) mod_dd_polynomial::get_dd_polynomial ( double precision, dimension(:), intent(in) derivatives )

Get the polynomial associated to a DD.

Returns: the array (of the same size) of the associated coefficients