gjp_common Module Reference

This module provides utility functions for computing Jacobi matrices and zeroth moments. More...

Functions/Subroutines
type(gjp_sparse_matrix) function	jacobi_matrix (n, alpha, beta)
	Computes the Jacobi matrix for given parameters. More...
real(dp) function	jacobi_zeroeth_moment (alpha, beta)
	Computes the zeroth moment for Jacobi polynomials. More...

Detailed Description

This module provides utility functions for computing Jacobi matrices and zeroth moments.

This module contains essential functions for working with Jacobi polynomials. These functions are particularly useful for various numerical methods that utilize Jacobi polynomials. The module provides a function for generating a Jacobi matrix and another for calculating the zeroth moment of Jacobi polynomials.

Functions included are:

• jacobi_matrix: Generates a Jacobi matrix based on given alpha and beta parameters.
• jacobi_zeroeth_moment: Calculates the zeroth moment of Jacobi polynomials.

Function/Subroutine Documentation

jacobi_matrix()

type(gjp_sparse_matrix) function gjp_common::jacobi_matrix	(	integer, intent(in)	n,
		real(dp), intent(in)	alpha,
		real(dp), intent(in)	beta
	)

Computes the Jacobi matrix for given parameters.

The Jacobi matrix is computed as: [ J_{i,j} = \begin{cases} \alpha & \text{if } i = j \ \beta & \text{if } |i-j| = 1 \ 0 & \text{otherwise} \end{cases} ]

Parameters

[in]	n	Size of the matrix, number of points
[in]	alpha	parameter for Jacobi polynomials
[in]	beta	parameter for Jacobi polynomials

Returns

A gjp_sparse_matrix representing the Jacobi matrix

Definition at line 49 of file gjp_common.f90.

    integer, intent(in) :: n ! Size of the matrix, number of points
    real(dp), intent(in) :: alpha, beta
    type(gjp_sparse_matrix) :: jacmat
    integer :: idx
    real(dp) :: ab, abi, a2b2
 
    allocate (jacmat%diagonal(n))
    allocate (jacmat%off_diagonal(n - 1))
 
    ab = alpha + beta
    abi = 2.0_dp + ab
    jacmat%diagonal(1) = (beta - alpha) / abi
    jacmat%off_diagonal(1) = 4.0_dp * (1.0_dp + alpha) * (1.0_dp + beta) &
                             / ((abi + 1.0_dp) * abi * abi)
    a2b2 = beta * beta - alpha * alpha
    do idx = 2, n
        abi = 2.0_dp * idx + ab
        jacmat%diagonal(idx) = a2b2 / ((abi - 2.0_dp) * abi)
        abi = abi**2
        if (idx < n) then
            jacmat%off_diagonal(idx) = 4.0_dp * idx * (idx + alpha) * (idx + beta) &
                                       * (idx + ab) / ((abi - 1.0_dp) * abi)
        end if
    end do
    jacmat%off_diagonal(1:n - 1) = sqrt(jacmat%off_diagonal(1:n - 1))

jacobi_zeroeth_moment()

real(dp) function gjp_common::jacobi_zeroeth_moment	(	real(dp), intent(in)	alpha,
		real(dp), intent(in)	beta
	)

Computes the zeroth moment for Jacobi polynomials.

The zeroth moment is computed using the formula: [ \text{zmom} = 2^{(\alpha + \beta + 1)} \frac{\Gamma(\alpha + 1) \Gamma(\beta + 1)}{\Gamma(2 + \alpha + \beta)} ] Where (\Gamma) is the gamma function.

Parameters

[in]	alpha	parameter for Jacobi polynomials
[in]	beta	parameter for Jacobi polynomials

Returns

The zeroth moment value

Note

The zeroth moment should always be positive

Definition at line 89 of file gjp_common.f90.

    real(dp), intent(in) :: alpha, beta
    real(dp) :: zmom
    real(dp) :: ab, abi
 
    ab = alpha + beta
    abi = 2.0_dp + ab
 
    zmom = 2.0_dp**(alpha + beta + 1.0_dp) * exp(log_gamma(alpha + 1.0_dp) &
                                                 + log_gamma(beta + 1.0_dp) - log_gamma(abi))
 
    if (zmom <= 0.0_dp) then
        error stop "Zeroth moment is not positive but should be"
    end if