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 algebraic form as described below, whose Cauchy Principal Value is to be computed. More...

## Public Member Functions

procedure get => getIntCauchy2

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) csnot

real(RKH), dimension(2) Pole

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 algebraic form as described below, whose Cauchy Principal Value is to be computed.

The full integrand is defined as,

$$f(x) = \frac{1}{(x - \ms{cs1})(x - \ms{cs2})} ~,~ x \in (-\infty \leq \ms{lb} < \min(\ms{cs1},\ms{cs2}), \min(\ms{cs1},\ms{cs2}) < \ms{ub} < \max(\ms{cs1},\ms{cs2})) ~\vee~ x \in (\min(\ms{cs1},\ms{cs2}) < \ms{lb} < \max(\ms{cs1},\ms{cs2}), \max(\ms{cs1},\ms{cs2}) < \ms{ub} \leq +\infty)$$

Depending on the choice of integration range, the integrand has either $$\ms{cs1}$$ or $$\ms{cs2}$$ as its Cauchy singularity (but not both).
The Cauchy Principal value of the integrand is,

$$\bigg[ \frac{\log(x - \ms{cs1}) - \log(x - \ms{cs2})}{\ms{cs1} - \ms{cs2}} \bigg]_{\ms{lb}}^{\ms{ub}} ~.$$

Parameters
 [in] lb : The input scalar of type real of kind RKH, containing the lower limit of integration. (optional, default = -2.) [in] ub : The input scalar of the same type and kind as lb, containing the upper limit of integration. (optional, default = +2.) [in] cs1 : The input scalar of the same type and kind as lb, containing the first pole (Cauchy singularity) of the integrand. Note that cs1 < cs2 must hold. (optional, default = -2.) [in] cs2 : The input scalar of the same type and kind as lb, containing the second pole (Cauchy singularity) of the integrand. Note that cs1 < cs2 must hold. (optional, default = +3.)

Possible calling interfaces

type(intCauchy2_type) :: integrand
integrand = intCauchy2_type(lb = lb, ub = ub, cs1 = cs1, cs2 = cs2)
print *, "description: ", integrand%desc
print *, "lower limit: ", integrand%lb
print *, "upper limit: ", integrand%ub
print *, "singularity: ", integrand%cs
print *, "Example integrand value: ", integrand%get(x)
print *, "Example integrand value without the Cauchy weight: ", integrand%getWeighted(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 algebraic form as described bel...
Warning
The two Cauchy singularities of the integrand must not be simultaneously present in the in the integration range.
This condition is verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.
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 1760 of file pm_quadTest.F90.

## ◆ get()

Definition at line 1763 of file pm_quadTest.F90.

References pm_kind::RKH.