mod_math_legendre Module Reference

Legendre integration. More...

Functions/Subroutines
subroutine	evaluate_legendre_points_and_weights (points, weights)
subroutine	evaluate_zero_legendre_and_derivative (n, legendre0, derivative0)
	Evaluate the Legendre polynomial of degree n and its derivative at the point 0.
subroutine	compute_legendre_first_root_and_derivative (n, legendre0, root, derivative)
	Compute the first positive root of Legendre polynomial of degree n and its derivative at 0.
double precision function	legendre_get_next_root_approximation (n, root)

Detailed Description

Legendre integration.

Function/Subroutine Documentation

compute_legendre_first_root_and_derivative()

subroutine mod_math_legendre::compute_legendre_first_root_and_derivative	(	integer, intent(in)	n,
		double precision, intent(in)	legendre0,
		double precision, intent(out)	root,
		double precision, intent(out)	derivative )

Compute the first positive root of Legendre polynomial of degree n and its derivative at 0.

Remarks:

• Odd Legendre polynomials have a root at 0 and the first derivative is known. This routine concerns even Legendre polynomials.
• Even Legendre polynomials have their first derivative equal to zero at 0.

The article of Glaser et al. proposes a specific algorithm to find the first positive root of the Legendre polynomials.

evaluate_legendre_points_and_weights()

subroutine mod_math_legendre::evaluate_legendre_points_and_weights	(	double precision, dimension(:)	points,
		double precision, dimension(:)	weights )

evaluate_zero_legendre_and_derivative()

subroutine mod_math_legendre::evaluate_zero_legendre_and_derivative	(	integer, intent(in)	n,
		double precision, intent(out)	legendre0,
		double precision, intent(out)	derivative0 )

Evaluate the Legendre polynomial of degree n and its derivative at the point 0.

Legendre polynomials Pn of degree n verifies the Bonnet recurrence formula:

(n+1) Pn+1(x) = (2n+1) x Pn(x) - Pn-1(x)

with P0(x) = 1 and P1(x) = x

For x = 0, we get

          n-1
Pn(0) = - --- Pn-2(0)
           n

The derivative verifies the following recurrence formula:

           (2n+1) Pn(x) - n P'n-1(x)
P'n+1(x) = ---------------------------
                      n+1

legendre_get_next_root_approximation()

double precision function mod_math_legendre::legendre_get_next_root_approximation	(	integer, intent(in)	n,
		double precision, intent(in)	root )