multidip_romberg Module Reference

Romberg quadrature for numerical integration. More...

Functions/Subroutines
subroutine	nested_cgreen_correct_romberg (z, a, b, c, n, sa, sb, m, l, k, v)
	Numerically correct asymptotic approximation of the Coulomb-Green integral (Romberg integration)
complex(wp) function	nested_cgreen_eval (z, c, n, sa, sb, m, l, k, rs, asy)
	Evaluate the Coulomb-Green's integrand.

Detailed Description

Romberg quadrature for numerical integration.

Author

J Benda

Date

2021

Function/Subroutine Documentation

nested_cgreen_correct_romberg()

subroutine multidip_romberg::nested_cgreen_correct_romberg	(	real(wp), intent(in)	z,
		real(wp), intent(in)	a,
		real(wp), intent(in)	b,
		real(wp), intent(in)	c,
		integer, intent(in)	n,
		integer, intent(in)	sa,
		integer, intent(in)	sb,
		integer, dimension(:), intent(in)	m,
		integer, dimension(:), intent(in)	l,
		complex(wp), dimension(:), intent(in)	k,
		complex(wp), intent(inout)	v )

Numerically correct asymptotic approximation of the Coulomb-Green integral (Romberg integration)

Author

J Benda

Date

2021 - 2023

This function computes the integral of the Coulomb-Green's integrand over Q = (a,+∞)^N \ (b,+∞)^N numerically. Currently, Romberg integration based on the trapezoidal rule is used.

Parameters

Z	Residual ion charge
a	Lower bound
b	Upper bound
c	Damping factor (additional exp(-c*r) added to all r^m functions)
N	Dimension (number of integration variables)
sa	Sign of the right-most Coulomb-Hankel function
sb	Sign of the left-most Coulomb-Hankel function
m	Array of integer powers of length N
l	Array of angular momenta of length N + 1
k	Array of linear momenta of length N + 1
v	Value of the asymptotic integral to correct (updated on exit from this subroutine)

Warning

This is currently only implemented for one-dimensional case containing no Green's function at all.

Todo

Generalize for arbitrary dimension!

Definition at line 58 of file multidip_romberg.f90.

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nested_cgreen_eval()

complex(wp) function multidip_romberg::nested_cgreen_eval	(	real(wp), intent(in)	z,
		real(wp), intent(in)	c,
		integer, intent(in)	n,
		integer, intent(in)	sa,
		integer, intent(in)	sb,
		integer, dimension(:), intent(in)	m,
		integer, dimension(:), intent(in)	l,
		complex(wp), dimension(:), intent(in)	k,
		real(wp), dimension(:), intent(in)	rs,
		logical, intent(in)	asy )

Evaluate the Coulomb-Green's integrand.

Author

J Benda

Date

2021 - 2024

Parameters

Z	Residual ion charge
c	Damping factor (additional exp(-c*r) added to all r^m functions)
N	Dimension (number of integration variables)
sa	Sign of the right-most Coulomb-Hankel function
sb	Sign of the left-most Coulomb-Hankel function
m	Array of integer powers of length N
l	Array of angular momenta of length N + 1
k	Array of linear momenta of length N + 1
rs	Single point in the multidimensional position space where the evaluate the integrand
asy	Use the asymptotic form of the Coulomb-Green's functions.

Warning

This is currently only implemented for one-dimensional case containing no Green's function at all.

Todo

Generalize to arbitrary dimension!

Definition at line 142 of file multidip_romberg.f90.

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