ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
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

logVolEll = getLogVolEll(gramian(1:ndim, 1:ndim))
Generate and return the natural logarithm of the volume of an -dimensional ellipsoid.
This module contains classes and procedures for setting up and computing the properties of the hyper-...
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

1program example
2
3 use pm_kind, only: SK, IK, LK
4 use pm_io, only: display_type
6
7 implicit none
8
9 real, allocatable :: gramian(:,:)
10
11 type(display_type) :: disp
12 disp = display_type(file = "main.out.F90")
13
14 call disp%skip()
15 call disp%show("gramian = reshape([1., 0., 0., 1.], [2,2]) ! correlation 0.")
16 gramian = reshape([1., 0., 0., 1.], [2,2])
17 call disp%show("exp(getLogVolEll(gramian)) ! correlation 0.")
18 call disp%show( exp(getLogVolEll(gramian)) )
19 call disp%skip()
20
21 call disp%skip()
22 call disp%show("gramian = reshape([1., 0.5, 0.5, 1.], [2,2]) ! correlation 0.5")
23 gramian = reshape([1., 0.5, 0.5, 1.], [2,2])
24 call disp%show("exp(getLogVolEll(gramian))")
25 call disp%show( exp(getLogVolEll(gramian)) )
26 call disp%skip()
27
28 call disp%skip()
29 call disp%show("gramian = reshape([1., 0.99, 0.99, 1.], [2,2]) ! correlation 0.99")
30 gramian = reshape([1., 0.99, 0.99, 1.], [2,2])
31 call disp%show("exp(getLogVolEll(gramian))")
32 call disp%show( exp(getLogVolEll(gramian)) )
33 call disp%skip()
34
35end program example
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example output
1
2gramian = reshape([1., 0., 0., 1.], [2,2]) ! correlation 0.
3exp(getLogVolEll(gramian)) ! correlation 0.
4+3.14159298
5
6
7gramian = reshape([1., 0.5, 0.5, 1.], [2,2]) ! correlation 0.5
8exp(getLogVolEll(gramian))
9+2.72069931
10
11
12gramian = reshape([1., 0.99, 0.99, 1.], [2,2]) ! correlation 0.99
13exp(getLogVolEll(gramian))
14+0.443175852
15
16
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


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.
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.
  2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

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.


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