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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See 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 |
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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 |
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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... | |
This is the derived type for generating test integrand objects of the following algebraic form.
The full integrand is defined as,
\begin{equation} \large f(x) = \left(\frac{x}{\ms{lb}} \right)^\alpha \exp\left( -\beta \left[ x - \ms{lb} \right] \right) ~, \ms{lb} \in (0, +\infty) \end{equation}
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 \(Q(\cdot)\):
\begin{equation} \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}) ~. \end{equation}
Otherwise, the integrand must be computed numerically, in which case, the integrand
component of object (representing the truth) is set to NaN
.
[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 ⛓
0 < lb
must hold for the corresponding input arguments.lb < ub
must hold for the corresponding input arguments.0 < beta
must hold for the corresponding input arguments.
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.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Definition at line 1025 of file pm_quadTest.F90.
procedure pm_quadTest::intGamUpp_type::get |
Definition at line 1029 of file pm_quadTest.F90.
References pm_kind::RKH.
real(RKH) pm_quadTest::intGamUpp_type::alpha |
Definition at line 1026 of file pm_quadTest.F90.
real(RKH) pm_quadTest::intGamUpp_type::beta |
Definition at line 1026 of file pm_quadTest.F90.
real(RKH) pm_quadTest::intGamUpp_type::normfac |
Definition at line 1027 of file pm_quadTest.F90.