math_utilities_mod – the SCATTERING code

This module holds 4 types of functions:

(1) algebraic functions: "percival_seaton_coefficient",

(2) geometric functions: "triangle_inequality_holds", "is_sum_even", "zero_projections_3j_condition"

(3) bessel functions: "riccati_bessel_j", "riccati_bessel_y" and "modified_bessel_k_ratio" that call functions from special_functions.f90 library

(4) interpolation procedures: "spline" and "ispline" functions

Uses

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Used by

Help
Graph Key

Nodes of different colours represent the following:

Solid arrows point from a submodule to the (sub)module which it is descended from. Dashed arrows point from a module or program unit to modules which it uses. Where possible, edges connecting nodes are given different colours to make them easier to distinguish in large graphs.

Functions

is_sum_even ispline percival_seaton_coefficient rgamma triangle_inequality_holds zero_projections_3j_condition

Subroutines

modified_bessel_k_ratio modified_bessel_temme_algorithm riccati_bessel_j riccati_bessel_y spline

Functions

public function is_sum_even(x, y, z) result(sum_even)

checks if the sum of 3 integers is an even integer

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	x	variables to check if the sum is even
integer(kind=int32),	intent(in)	::	y	variables to check if the sum is even
integer(kind=int32),	intent(in)	::	z	variables to check if the sum is even

Return Value logical

(out) result: true/false

public function ispline(u_, N_, x_, y_, b_, c_, d_) result(spl_result)

returns interpolated value at guven u_ point number of points and ascending order of x is not checked since ispline is called after "spline" where these checks are done

Arguments

Type	Intent		Name
real(kind=dp),	intent(in)	::	u_	point at which the tabulated value is interpolated
integer(kind=int32),	intent(in)	::	N_	number of grid points
real(kind=dp),	intent(in)	::	x_(N_)	grid points
real(kind=dp),	intent(in)	::	y_(N_)	tabulated values
real(kind=dp),	intent(in)	::	b_(N_)	arrays with coefficients of the spline function
real(kind=dp),	intent(in)	::	c_(N_)	arrays with coefficients of the spline function
real(kind=dp),	intent(in)	::	d_(N_)	arrays with coefficients of the spline function

Return Value real(kind=dp)

interpolated value at u_

public function percival_seaton_coefficient(j_, j_prime_, lambda_, omega_) result(percival_seaton_coefficient_)

calculates Percival-Seaton coefficients (body-fixed variant) $\begin{equation} \label{eq:algebraic_coeffs} g_{{\lambda},\gamma,\gamma'}^{Jp} = \delta_{\bar{\Omega},\bar{\Omega}'} (-1)^{\bar{\Omega}} \sqrt{(2j+1)(2j'+1)} \begin{pmatrix} j & j' & \lambda \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} j & j' & \lambda \\ \bar{\Omega} & -\bar{\Omega} & 0 \end{pmatrix}. \end{equation}$

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	j_	pre-collisional rotational angular momentum
integer(kind=int32),	intent(in)	::	j_prime_	post-collisional rotational angular momentum
integer(kind=int32),	intent(in)	::	lambda_	Legendre expansion coefficient $ \lambda$
integer(kind=int32),	intent(in)	::	omega_	$\bar{\Omega}$

Return Value real(kind=dp)

(out) result: percival seaton coefficient in the body-fixed frame

public function rgamma(x, odd, even) result(rgamma_val)

Calculates 1/Gamma(1-X); modernized version of Molscat's rgamma function; see: https://github.com/molscat/molscat/blob/36fa8f93a92f851e9d84245dd6a972e2910541c5/source_code/rbesjy.f --------------------------------------------------------------------!

Arguments

Type	Intent		Name
real(kind=dp),	intent(in)	::	x
real(kind=dp),	intent(out)	::	odd
real(kind=dp),	intent(out)	::	even

Return Value real(kind=dp)

public function triangle_inequality_holds(x, y, z) result(holds)

check if the triangle inequality for 3 variables hols

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	x	variables to check the triangle inequality
integer(kind=int32),	intent(in)	::	y	variables to check the triangle inequality
integer(kind=int32),	intent(in)	::	z	variables to check the triangle inequality

Return Value logical

(out) result: true/false

public function zero_projections_3j_condition(x, y, z) result(is_valid)

checks the condition for nonvanishing 3-j symbol with zero projections: triangle inequality on x,y,z and if the sum x+y+z is an even integer

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	x	variables to check for 3-j symbol conditions
integer(kind=int32),	intent(in)	::	y	variables to check for 3-j symbol conditions
integer(kind=int32),	intent(in)	::	z	variables to check for 3-j symbol conditions

Return Value logical

(out) result: true/false if conditions are met

Subroutines

public subroutine modified_bessel_k_ratio(l_, x_, ratio_)

calculates the ratio of the modified Bessel function of the second kind K_{l_ + 1/2}(x) and its first derivative (Eq. 8 in the "Solution of the coupled equations" section) Uses Temme's algorithm [N. M. Temme, J. Comput. Phys. 19 (1975) 324], implemented in "modified_bessel_temme_algorithm" subroutine; Unfortunately, the "ikv" function from special_functions library failed at large x_ values.

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	l_	l - order of the function (without the 1/2 factor!)
real(kind=dp),	intent(in)	::	x_	x - argument of the function
real(kind=dp),	intent(inout)	::	ratio_	ratio of the modified Bessel function of the second kind to its derivative

public subroutine modified_bessel_temme_algorithm(v, x, ck, dk, ek)

Implementation of the Temme's algorithm [N. M. Temme, J. Comput. Phys. 19 (1975) 324] to calculating modified Bessel functions of the second kind. This is a direct modernization of the "mbessk" subroutine in MOLSCAT: https://github.com/molscat/molscat/blob/master/source_code/rbessk.f

Arguments

Type	Intent		Name
real(kind=dp),	intent(in)	::	v
real(kind=dp),	intent(in)	::	x
real(kind=dp),	intent(out)	::	ck
real(kind=dp),	intent(out)	::	dk
real(kind=dp),	intent(out)	::	ek

public subroutine riccati_bessel_j(l_, x_, j_, jp_)

calculates the Riccati-Bessel function of the first kind and its first derivative. Calls the rctj function from special_functions.f90

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	l_	l - order of the Riccati-Bessel function of the first kind
real(kind=dp),	intent(in)	::	x_	x - argument of the Riccati-Bessel function of the first kind
real(kind=dp),	intent(inout)	::	j_	j_{l} (x) - Riccati-Bessel funciton of the first kind
real(kind=dp),	intent(inout)	::	jp_	j_{l}' (x) - derivative of the Riccati-Bessel funciton of the first kind

public subroutine riccati_bessel_y(l_, x_, y_, yp_)

calculates the Riccati-Bessel function of the second kind and its first derivative. Calls the rcty function from special_functions.f90

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	l_	l - order of the Riccati-Bessel function of the second kind
real(kind=dp),	intent(in)	::	x_	x - argument of the Riccati-Bessel function of the second kind
real(kind=dp),	intent(inout)	::	y_	y_{l} (x) - Riccati-Bessel funciton of the second kind
real(kind=dp),	intent(inout)	::	yp_	y_{l}' (x) - derivative of the Riccati-Bessel funciton of the second kind

public subroutine spline(N_, x_, y_, b_, c_, d_)

determines b, c and d coefficients of the cubic spline function y(x) = y_i + b_i * dx + c_i * dx^2 + d_i * dx^3, where dx = x - x_i, and x_i <= x < x_i+1. The algorithm is based on Gerald, C., and Wheatley, P., "Applied Numerical Analysis", Addison-Wesley, 1994.

Arguments

Type	Intent		Name
integer(kind=int32),	intent(in)	::	N_	number of grid points
real(kind=dp),	intent(in)	::	x_(N_)	grid points (ascending order)
real(kind=dp),	intent(in)	::	y_(N_)	tabulated values
real(kind=dp),	intent(out)	::	b_(N_)	arrays with coefficients of the spline function
real(kind=dp),	intent(out)	::	c_(N_)	arrays with coefficients of the spline function
real(kind=dp),	intent(out)	::	d_(N_)	arrays with coefficients of the spline function

math_utilities_mod Module

Contents

Functions

Subroutines

Uses

Used by

Contents

Functions

Subroutines

Functions

public function is_sum_even(x, y, z) result(sum_even)

Arguments

Return Value logical

public function ispline(u_, N_, x_, y_, b_, c_, d_) result(spl_result)

Arguments

Return Value real(kind=dp)

public function percival_seaton_coefficient(j_, j_prime_, lambda_, omega_) result(percival_seaton_coefficient_)

Arguments

Return Value real(kind=dp)

public function rgamma(x, odd, even) result(rgamma_val)

Arguments

Return Value real(kind=dp)

public function triangle_inequality_holds(x, y, z) result(holds)

Arguments

Return Value logical

public function zero_projections_3j_condition(x, y, z) result(is_valid)

Arguments

Return Value logical

Subroutines

public subroutine modified_bessel_k_ratio(l_, x_, ratio_)

Arguments

public subroutine modified_bessel_temme_algorithm(v, x, ck, dk, ek)

Arguments

public subroutine riccati_bessel_j(l_, x_, j_, jp_)

Arguments

public subroutine riccati_bessel_y(l_, x_, y_, yp_)

Arguments

public subroutine spline(N_, x_, y_, b_, c_, d_)

Arguments