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 |
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 |
Public Attributes inherited from pm_quadTest::integrand_type | |
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 ⛓
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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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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.