ParaMonte Fortran 2.0.0 Parallel Monte Carlo and Machine Learning LibrarySee the latest version documentation.

This is the derived type for generating test integrand objects of the following algebraic form. More...

## Public Member Functions

procedure get => getIntGamUpp

Public Member Functions inherited from pm_quadTest::integrand_type
procedure(get_proc), deferred get
The function member returning the value of the unweighted integrand (whether Cauchy/sin/cos/algebraically types of weights) at a specified input point x. More...

## Public Attributes

real(RKH) alpha

real(RKH) beta

real(RKH) normfac

real(RKH) lb
The scalar of type real of the highest kind supported by the library RKH, containing the lower limit of integration. More...

real(RKH) ub
The scalar of type real of the highest kind supported by the library RKH, containing the upper limit of integration. More...

real(RKH) integral
The scalar of type real of the highest kind supported by the library RKH, containing the true result of integration. More...

real(RKH), dimension(:), allocatable break
The scalar of type real of the highest kind supported by the library RKH, containing the points of difficulties of integration. More...

type(wcauchy_type), allocatable wcauchy
The scalar of type wcauchy_type, containing the Cauchy singularity of the integrand. More...

character(:, SK), allocatable desc
The scalar allocatable character of default kind SK containing a description of the integrand and integration limits and difficulties. More...

## Detailed Description

This is the derived type for generating test integrand objects of the following algebraic form.

The full integrand is defined as,

$$\large f(x) = \left(\frac{x}{\ms{lb}} \right)^\alpha \exp\left( -\beta \left[ x - \ms{lb} \right] \right) ~, \ms{lb} \in (0, +\infty)$$

where $$\beta > 0$$ with integration range as $$[\ms{lb}, \ms{ub}]$$ where $$\ms{lb} < \ms{ub} < +\infty$$.
The integrand has a singularity at $$x = 0$$ with $$\alpha < 0$$, but the $$\ms{lb}$$ range does not allow singularity to enter the integrand.
When $$\alpha > -1, \ms{ub} = +\infty$$, this integral can be computed via regularized upper incomplete Gamma function $$Q(\cdot)$$:

$$\large f(x) = \frac{ \exp(\beta \ms{lb}) }{ \ms{lb}^\alpha ~ \beta^{\alpha + 1} } ~ \Gamma(\alpha + 1) ~ Q(\alpha + 1, \beta \ms{lb}) - \frac{ \exp(\beta \ms{ub}) }{ \ms{ub}^\alpha ~ \beta^{\alpha + 1} } ~ \Gamma(\alpha + 1) ~ Q(\alpha + 1, \beta \ms{ub}) ~.$$

Otherwise, the integrand must be computed numerically, in which case, the integrand component of object (representing the truth) is set to NaN.

Parameters
 [in] lb : The input scalar of type real of kind RKH. (optional, default = 1.) [in] ub : The input scalar of the same type and kind as a. (optional, default = getInfPos(self%ub) [in] alpha : The input scalar of type integer of default kind IK, standing for Lower Factor, such that lb = lf * pi is the lower bound of integration. (optional, default = +1.) [in] beta : The input scalar of type integer of default kind IK, standing for Upper Factor, such that ub = uf * pi is the upper bound of integration. (optional, default = +1.)

Possible calling interfaces

type(intGamUpp_type) :: integrand
integrand = intGamUpp_type()
print *, "description: ", integrand%desc
print *, "lower limit: ", integrand%alpha
print *, "lower limit: ", integrand%beta
print *, "lower limit: ", integrand%lb
print *, "upper limit: ", integrand%ub
print *, "Example integrand value: ", integrand%get(x)
This module contains a collection of interesting or challenging integrands for testing or examining t...
This is the derived type for generating test integrand objects of the following algebraic form.
Warning
The condition 0 < lb must hold for the corresponding input arguments.
The condition lb < ub must hold for the corresponding input arguments.
The condition 0 < beta must hold for the corresponding input arguments.
integrand_type
Test:

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
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Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 1025 of file pm_quadTest.F90.

## ◆ get()

Definition at line 1029 of file pm_quadTest.F90.

References pm_kind::RKH.

## ◆ alpha

Definition at line 1026 of file pm_quadTest.F90.