Fortran Routines¶

This is the documentation of the Fortran routines for calculating the Wright function. Unlike their counterpart in C, they use ordinary arithmetic with different types of precision (32,64,128) for the calculation.

We have implemented both the scalar version of the function, and the vectorized counterparts with respect to the \(x\) variable. They are all private inside the module, and are accesible from a unique interface called wright.

Module

Description

Module containg the implementation of the routines for computing the Wright function on the real line.

Quick access

Variables: d_pi, done, donei, s_pi, sone, sonei, t_pi, tone, tonei
Routines: dwright(), dwright_vec(), swright(), swright_vec(), twright(), twright_vec()

Needed modules

iso_fortran_env (real32(), real64(), real128(), error_unit())
brentmod (fminbnd()): This module contains the implementation of Brent’s algorithm for the minimization of a function \(f(x)\) in the interval \([a,b]\).

Variables

wrightmod/d_pi [real,private/parameter/optional/default=3.141592653589793_real64]¶
wrightmod/done [real,private/parameter/optional/default=1.0_real64]¶
wrightmod/donei [complex,private/parameter/optional/default=(0.0,1.0_real64)]¶
wrightmod/s_pi [real,private/parameter/optional/default=3.14159265_real32]¶
wrightmod/sone [real,private/parameter/optional/default=1.0_real32]¶
wrightmod/sonei [complex,private/parameter/optional/default=(0.0,1.0_real32)]¶
wrightmod/t_pi [real,private/parameter/optional/default=3.1415926535897932384626433832795_real128]¶
wrightmod/tone [real,private/parameter/optional/default=1.0_real128]¶
wrightmod/tonei [complex,private/parameter/optional/default=(0.0,1.0_real128)]¶

Subroutines and functions

function wrightmod/dwright(x, t, lambda, mu[, n])¶

Parameters

x [real] :: The x in the argument \(-|x|t^\lambda\)
t [real] :: The t in the argument \(-|x|t^\lambda\) ( \(t > 0\) )
lambda [real] :: First parameter of the Wright function
mu [real] :: Second parameter of the Wright function
n [integer] :: Optional, number of quadrature points

Return

w [real] :: \(t^{\mu - 1} W_{\lambda,\mu} (-|x|t^{\lambda})\)

function wrightmod/swright(x, t, lambda, mu[, n])¶

Parameters

x [real] :: The x in the argument \(-|x|t^\lambda\)
t [real] :: The t in the argument \(-|x|t^\lambda\) ( \(t > 0\) )
lambda [real] :: First parameter of the Wright function
mu [real] :: Second parameter of the Wright function
n [integer] :: Optional, number of quadrature points

Return

w [real] :: \(t^{\mu - 1} W_{\lambda,\mu} (-|x|t^{\lambda})\)

function wrightmod/twright(x, t, lambda, mu[, n])¶

Parameters

x [real] :: The x in the argument \(-|x|t^\lambda\)
t [real] :: The t in the argument \(-|x|t^\lambda\) ( \(t > 0\) )
lambda [real] :: First parameter of the Wright function
mu [real] :: Second parameter of the Wright function
n [integer] :: Optional, number of quadrature points

Return

w [real] :: \(t^{\mu - 1} W_{\lambda,\mu} (-|x|t^{\lambda})\)

function wrightmod/dwright_vec(x, t, lambda, mu[, n])¶

Parameters

x (*) [real] :: The x in the argument \(-|x|t^\lambda\)
t [real] :: The t in the argument \(-|x|t^\lambda\) ( \(t > 0\) )
lambda [real] :: First parameter of the Wright function
mu [real] :: Second parameter of the Wright function
n [integer] :: Optional to arbitrarly set the number of quadrature points

Return

w (*) [real,allocatable] :: \(t^{\mu - 1} W_{\lambda,\mu} (-|x|t^{\lambda})\)

function wrightmod/swright_vec(x, t, lambda, mu[, n])¶

Parameters

x (*) [real] :: The x in the argument \(-|x|t^\lambda\)
t [real] :: The t in the argument \(-|x|t^\lambda\) ( \(t > 0\) )
lambda [real] :: First parameter of the Wright function
mu [real] :: Second parameter of the Wright function
n [integer] :: Optional to arbitrarly set the number of quadrature points

Return

w (*) [real,allocatable] :: \(t^{\mu - 1} W_{\lambda,\mu} (-|x|t^{\lambda})\)

function wrightmod/twright_vec(x, t, lambda, mu[, n])¶

Parameters

x (*) [real] :: The x in the argument \(-|x|t^\lambda\)
t [real] :: The t in the argument \(-|x|t^\lambda\) ( \(t > 0\) )
lambda [real] :: First parameter of the Wright function
mu [real] :: Second parameter of the Wright function
n [integer] :: Optional to arbitrarly set the number of quadrature points

Return

w (*) [real,allocatable] :: \(t^{\mu - 1} W_{\lambda,\mu} (-|x|t^{\lambda})\)

Brent’s algorithm for constrained minimization¶

To compute the \(\xi\) parameter needed for the parabolic contour we need to solve a box-constrained optimization problem. For this task we use Brent’s algorithm for optimization without derivatives.

Module

Description

This module contains the implementation of Brent’s algorithm for the minimization of a function \(f(x)\) in the interval \([a,b]\).

The implementation is the one from:

Note

Original FORTRAN77 version by Richard Brent. FORTRAN90 version by John Burkardt.

See: https://people.math.sc.edu/Burkardt/f_src/brent/brent.html

Quick access