tensor2D.H

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License
    This file is part of OpenFOAM.
 
    OpenFOAM is free software: you can redistribute it and/or modify it
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Typedef
    Foam::tensor2D
 
Description
    Tensor2D of scalars, i.e. Tensor2D<scalar>.
 
    Analytical functions for the computation of complex eigenvalues and
    complex eigenvectors from a given tensor2D.
 
    Reference:
    \verbatim
        2-by-2 eigenvalue algorithm (tags:F; B):
            Ford, W. (2014).
            Numerical linear algebra with applications: Using MATLAB.
            London: Elsevier/Academic Press.
            DOI:10.1016/C2011-0-07533-6
 
            Blinn, J. (1996).
            Consider the lowly 2 x 2 matrix.
            IEEE Computer Graphics and Applications, 16(2), 82-88.
            DOI:10.1109/38.486688
 
        2-by-2 eigenvector algorithm (tag:K):
            Knill, O. (2004).
            Mathematics Math21b Fall 2004.
            bit.ly/2kjPVlX (Retrieved:06-09-19)
 
        Kahan summation algorithm for 2-by-2 matrix determinants (tag:JLM):
            Jeannerod, C.-P., Louvet, N., & Muller, J.-M., (2013).
            Further analysis of Kahan's algorithm for the accurate computation
            of 2x2 determinants.
            Math. Comp. 82 (2013), 2245-2264.
            DOI:10.1090/S0025-5718-2013-02679-8
    \endverbatim
 
See also
    Test-Tensor2D.C
 
SourceFiles
    tensor2D.C
 
\*---------------------------------------------------------------------------*/
 
#ifndef tensor2D_H
#define tensor2D_H
 
#include "Tensor2D.H"
#include "complex.H"
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
namespace Foam
{
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
typedef Tensor2D<scalar> tensor2D;
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
//- Return complex eigenvalues of a given tensor2D
//  \param T tensor2D
//
//  \return Vector2D<complex> eigenvalues
Vector2D<complex> eigenValues(const tensor2D& T);
 
 
//- Return a complex eigenvector corresponding to
//- a given complex eigenvalue of a given tensor2D
//  \param T tensor2D
//  \param eVal complex eigenvalue
//  \param standardBasis tensor2D orthogonal component, e.g. vector2D(1, 0)
//
//  \return Vector2D<complex> eigenvector
Vector2D<complex> eigenVector
(
    const tensor2D& T,
    const complex& eVal,
    const Vector2D<complex>& standardBasis
);
 
 
//- Return complex eigenvectors corresponding to
//- given complex eigenvalues of a given tensor2D
//  \param T tensor2D
//  \param eVals eigenvalues
//
//  \return Tensor2D<complex> eigenvectors, each row is an eigenvector
Tensor2D<complex> eigenVectors
(
    const tensor2D& T,
    const Vector2D<complex>& eVals
);
 
 
//- Return complex eigenvectors of a given tensor2D by computing
//- the complex eigenvalues of the tensor2D in the background
//  \param T tensor2D
//
//  \return Tensor2D<complex> eigenvectors, each row is an eigenvector
Tensor2D<complex> eigenVectors(const tensor2D& T);
 
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
} // End namespace Foam
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
#include "sphericalTensor2D.H"
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
#endif
 
// ************************************************************************* //