pm_quadTest::intCauchy2_type Type Reference

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...

Inheritance diagram for pm_quadTest::intCauchy2_type:

Collaboration diagram for pm_quadTest::intCauchy2_type:

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
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...

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,

\begin{equation} 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) \end{equation}

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,

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

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 ⛓

use pm_quadTest, only: intCauchy2_type
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)

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.

See also

integrand_type

Test:

test_pm_quadPack

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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Copyright

Computational Data Science Lab

Author:

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

Definition at line 1760 of file pm_quadTest.F90.

Member Function/Subroutine Documentation

get()

procedure pm_quadTest::intCauchy2_type::get

Definition at line 1763 of file pm_quadTest.F90.

References pm_kind::RKH.

Member Data Documentation

csnot

real(RKH) pm_quadTest::intCauchy2_type::csnot

Definition at line 1761 of file pm_quadTest.F90.

Pole

real(RKH), dimension(2) pm_quadTest::intCauchy2_type::Pole

Definition at line 1761 of file pm_quadTest.F90.

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The documentation for this type was generated from the following file:

• src/fortran/main/pm_quadTest.F90