pm_ellipsoid::getLogVolEll Interface Reference

Generate and return the natural logarithm of the volume of an \(\ndim\)-dimensional ellipsoid.
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Detailed Description

Generate and return the natural logarithm of the volume of an \(\ndim\)-dimensional ellipsoid.

See the documentation of pm_ellipsoid for computational and algorithmic details.

Parameters

[in]

gramian

: The input matrix of the same type and kind as the output logVolEll, containing the upper triangle and diagonal of the representative Gramian matrix of the ellipsoid.

Returns

logVolEll : The output scalar of,

1. type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),

containing natural logarithm of the volume of the \(\ndim\)-dimensional hyper-ellipsoid.

Possible calling interfaces ⛓

use pm_ellipsoid, only: getLogVolEll
 
logVolEll = getLogVolEll(gramian(1:ndim, 1:ndim))

Warning

If the Cholesky factorization of the input Gramian fails, the procedures of this generic interface will abort the program by calling error stop.

The condition size(gramian, 1) == size(gramian, 2) must hold for the corresponding input arguments.
The condition \(0. < \left|\ms{gramian}\right|\) must hold for the corresponding input arguments.
In other words, the input Gramian must be a positive definite matrix.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.

Note

Computing the volume of an ellipsoid using its Gramian in a fixed dimension as implemented by the procedure of this generic interface are computationally costly.
The unnecessary costs can be eliminated by precomputing the natural logarithm of the volume of the unit ball in the desired dimension once via setLogVolUnitBall and adding to it the sum of the natural logarithms of the diagonal elements of the Cholesky factorization of the representative Gramian matrix of the ellipsoid.

See also

getLogVolEll

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, LK
    use pm_io, only: display_type
    use pm_ellipsoid, only: getLogVolEll
 
    implicit none
 
    real, allocatable :: gramian(:,:)
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("gramian = reshape([1., 0., 0., 1.], [2,2]) ! correlation 0.")
                    gramian = reshape([1., 0., 0., 1.], [2,2])
    call disp%show("exp(getLogVolEll(gramian)) ! correlation 0.")
    call disp%show( exp(getLogVolEll(gramian)) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("gramian = reshape([1., 0.5, 0.5, 1.], [2,2]) ! correlation 0.5")
                    gramian = reshape([1., 0.5, 0.5, 1.], [2,2])
    call disp%show("exp(getLogVolEll(gramian))")
    call disp%show( exp(getLogVolEll(gramian)) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("gramian = reshape([1., 0.99, 0.99, 1.], [2,2]) ! correlation 0.99")
                    gramian = reshape([1., 0.99, 0.99, 1.], [2,2])
    call disp%show("exp(getLogVolEll(gramian))")
    call disp%show( exp(getLogVolEll(gramian)) )
    call disp%skip()
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
gramian = reshape([1., 0., 0., 1.], [2,2]) ! correlation 0.
exp(getLogVolEll(gramian)) ! correlation 0.
+3.14159298
 
 
gramian = reshape([1., 0.5, 0.5, 1.], [2,2]) ! correlation 0.5
exp(getLogVolEll(gramian))
+2.72069931
 
 
gramian = reshape([1., 0.99, 0.99, 1.], [2,2]) ! correlation 0.99
exp(getLogVolEll(gramian))
+0.443175852
 
 

Test:

test_pm_ellipsoid

Todo:

High Priority: A positive-definiteness runtime check for the gramian input argument of this generic interface must be added.

Final Remarks ⛓

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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

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.

Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, April 23, 2017, 1:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin

Definition at line 746 of file pm_ellipsoid.F90.

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

• src/fortran/main/pm_ellipsoid.F90