ParaMonte Fortran 2.0.0 Parallel Monte Carlo and Machine Learning LibrarySee the latest version documentation.
pm_ellipsoid Module Reference

This module contains classes and procedures for setting up and computing the properties of the hyper-ellipsoids in arbitrary dimensions. More...

## Data Types

interface  getCountMemberEll
Generate and return the number of points that are members (i.e., inside) of the specified $$\ndim$$-dimensional ellipsoid.
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interface  getLogVolEll
Generate and return the natural logarithm of the volume of an $$\ndim$$-dimensional ellipsoid.
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interface  getLogVolUnitBall
Generate and return the natural logarithm of the volume of an $$\ndim$$-dimensional ball of unit-radius. More...

interface  isMemberEll
Generate and return .true. if and only if the input point is a member (i.e., inside) of the specified $$\ndim$$-dimensional ellipsoid.
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interface  setLogVolUnitBall
Return the natural logarithm of the volume of an $$\ndim$$-dimensional ball of unit-radius. More...

## Variables

character(*, SK), parameter MODULE_NAME = "@pm_ellipsoid"

## Detailed Description

This module contains classes and procedures for setting up and computing the properties of the hyper-ellipsoids in arbitrary dimensions.

An ellipsoid in Euclidean geomtery $$\ell$$ is defined by its representative Gramian matrix $$\Sigma_\ell$$, containing all points in $$\mathbb{R}^\ndim$$ that satisfy,

$$\large (X - \mu_\ell)^T ~ \Sigma_\ell^{-1} ~ (X - \mu_\ell) \leq 1 ~,$$

where $$\mu_\ell$$ represents the center of the ellipsoid, $$(X - \mu_\ell)^T$$ is the transpose of the vector $$(X - \mu_\ell)$$, and $$\Sigma_\ell^{-1}$$ is the inverse of the matrix $$\Sigma_\ell$$.

The volume of this ellipsoid is given by,

$$\large V(\ell) = V_\ndim \sqrt{\left| \Sigma_\ell \right|} ~,$$

where $$\left|\Sigma_\ell\right|$$ is the determinant of $$\Sigma_\ell$$ and,

$$\large V_\ndim = \frac{\pi^{\ndim / 2}}{\up\Gamma(1 + \ndim / 2)} = \begin{cases} \frac{1}{(\ndim/2)!} \pi^{\ndim/2} & \text{if \ndim is even} \\\\ 2^\ndim \frac{1}{\ndim!} \big( \frac{\ndim-1}{2} \big)! ~ \pi^{(\ndim-1)/2} & \text{if \ndim is odd} \end{cases}$$

is the volume of an $$\ndim$$-ball (that is, a unit-radius $$\ndim$$-dimensional hyper-ball).
It is readily seen that the corresponding unit-volume ellipsoid $$\widehat\ell$$ has the representative Gramian matrix,

$$\large \Sigma_{\widehat\ell} = V_\ell^{-2/\ndim} ~ \Sigma_\ell ~.$$

More generally, to scale an ellipsoid $$\ell$$ by some factor $$\alpha$$ along each coordinate axis, it suffices to be used the new scaled ellipsoid $$\ell^*$$ with the representative Gramian matrix,

$$\large \Sigma_{\ell^*} = \alpha^2 ~ \Sigma_\ell ~,$$

in which case, the volume of $$\ell^*$$ becomes,

$$\large V_{\ell^*} = \alpha^\ndim ~ V_\ell ~.$$

The surface area of an $$\ndim$$-ball
The surface area $$S$$ of an $$\ndim$$-dimensional ball of unit-radius is related to its volume $$V$$ as,

$$S_\ndim = \ndim \times V_\ndim ~,$$

where the natural logarithm of $$V$$ is returned by getLogVolUnitBall and setLogVolUnitBall.

Note
1. A ellipsoid with a diagonal representative Gramian matrix $$\Sigma_\ell$$ is an uncorrelated hyper-ellipsoid whose axes are parallel to the coordinate axes.
2. An uncorrelated hyper-ellipsoid with equal diagonal elements in its representative Gramian matrix $$\Sigma_\ell$$ is a hyper-ball.
pm_distUnifEll
pm_distUnifPar
pm_distMultiNorm
n-sphere
n-ball
Test:
test_pm_ellipsoid
Todo:
Critical Priority: The excluded procedure getLogVolUnion() in this module needs cleanup and merging with this module.

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.
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

## ◆ MODULE_NAME

 character(*, SK), parameter pm_ellipsoid::MODULE_NAME = "@pm_ellipsoid"

Definition at line 111 of file pm_ellipsoid.F90.