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  •  JAMA Pythagorean Theorem: a = 3 b = 4 r = sqrt(square(a) + square(b)) r = 5 r = sqrt(a^2 + b^2) without under/overflow
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    package JAMA Class to obtain eigenvalues and eigenvectors of a real matrix. If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal (i.e. A = V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the identity matrix). If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.times(V) equals V.times(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon V.cond().
    author Paul Meagher
    license PHP v3.0
    version 1.1

     Methods

    Constructor: Check for symmetry, then construct the eigenvalue decomposition

    __construct(\A $Arg) : \Structure
    access public

    Parameters

    $Arg

    \A

    Square matrix

    Returns

    \Structureto access D and V.

    Return the block diagonal eigenvalue matrix

    getD() : \D
    access public

    Returns

    \D

    Return the imaginary parts of the eigenvalues

    getImagEigenvalues() : \imag(diag(D))
    access public

    Returns

    \imag(diag(D))

    Return the real parts of the eigenvalues

    getRealEigenvalues() : \real(diag(D))
    access public

    Returns

    \real(diag(D))

    Return the eigenvector matrix

    getV() : \V
    access public

    Returns

    \V

    Performs complex division.

    cdiv($xr, $xi, $yr, $yi) 
    access private

    Parameters

    $xr

    $xi

    $yr

    $yi

    Nonsymmetric reduction from Hessenberg to real Schur form.

    hqr2() 

    Code is derived from the Algol procedure hqr2, by Martin and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutine in EISPACK.

    access private

    Nonsymmetric reduction to Hessenberg form.

    orthes() 

    This is derived from the Algol procedures orthes and ortran, by Martin and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutines in EISPACK.

    access private

    Symmetric tridiagonal QL algorithm.

    tql2() 

    This is derived from the Algol procedures tql2, by Bowdler, Martin, Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutine in EISPACK.

    access private

    Symmetric Householder reduction to tridiagonal form.

    tred2() 
    access private

     Properties

     

    $H : array
     

    $V : array
     

    $cdivi 
     

    $cdivr : float
     

    $d : array
     

    $e 
     

    $issymmetric : int
     

    $n : int
     

    $ort : array