LA::Eigenvalues Class Reference

Computes eigenvalues and eigenvectors of a real (non-complex) matrix. More...

#include <LA_Eigenvalues.h>

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[legend]List of all members.

Public Member Functions
	Eigenvalues (const Matrix &A)
	Check for symmetry, then construct the eigenvalue decomposition.
const Matrix &	getV () const
	Returns a reference to the matrix with the eigenvectors.
const Vector &	getRealEigenvalues () const
	Returns a reference to the vector with the real eigenvalues.
const Vector &	getImagEigenvalues () const
	Returns a reference to the vector with the real eigenvalues.
Matrix	getD () const
	Computes the block diagonal eigenvalue matrix.
Private Member Functions
	Eigenvalues (const Eigenvalues &ev)
Eigenvalues &	operator= (const Eigenvalues &ev)
void	tred2 ()
void	tql2 ()
void	orthes ()
void	cdiv (double xr, double xi, double yr, double yi)
void	hqr2 ()
Private Attributes
int	n
	Row and column dimension (square matrix).
int	issymmetric
Vector	d
	Arrays for internal storage of eigenvalues.
Vector	e
Matrix	V
	Array for internal storage of eigenvectors.
Matrix	H
	Array for internal storage of nonsymmetric Hessenberg form.
Vector	ort
	Working storage for nonsymmetric algorithm.
double	cdivr
double	cdivi

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Detailed Description

Computes eigenvalues and eigenvectors of a real (non-complex) matrix.

If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. That is, the diagonal values of D are the eigenvalues, and V*V' = I, where I is the identity matrix. The columns of V represent the eigenvectors in the sense that A*V = V*D.

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, a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex eigenvalues look like

        u + iv     .        .          .      .    .
          .      u - iv     .          .      .    .
          .        .      a + ib       .      .    .
          .        .        .        a - ib   .    .
          .        .        .          .      x    .
          .        .        .          .      .    y
 

then D looks like

          u        v        .          .      .    .
         -v        u        .          .      .    . 
          .        .        a          b      .    .
          .        .       -b          a      .    .
          .        .        .          .      x    .
          .        .        .          .      .    y
 

This keeps V a real matrix in both symmetric and non-symmetric cases, and A*V = V*D.

The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon the condition number of V.

(Adapted from JAMA, a Java Matrix Library, developed by jointly by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).

Definition at line 67 of file LA_Eigenvalues.h.

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Constructor & Destructor Documentation

LA::Eigenvalues::Eigenvalues (  const Matrix &  A  )

Check for symmetry, then construct the eigenvalue decomposition. Parameters: A  Square real (non-complex) matrix Definition at line 8 of file LA_Eigenvalues.cpp. References LA::Matrix::columns(), LA::Vector::create(), LA::Matrix::create(), e, H, hqr2(), issymmetric, ort, orthes(), tql2(), tred2(), and V.

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LA::Eigenvalues::Eigenvalues (  const Eigenvalues &  ev  )  [inline, private]

Definition at line 77 of file LA_Eigenvalues.h.

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Member Function Documentation

Eigenvalues& LA::Eigenvalues::operator= (  const Eigenvalues &  ev  )  [inline, private]

Definition at line 78 of file LA_Eigenvalues.h.

const Matrix & LA::Eigenvalues::getV (   )  const

Returns a reference to the matrix with the eigenvectors. Returns: Reference to a matrix with the eigenvectors. Definition at line 64 of file LA_Eigenvalues.cpp. References V.

const Vector & LA::Eigenvalues::getRealEigenvalues (   )  const

Returns a reference to the vector with the real eigenvalues. Returns: Reference to the vector with the real eigenvalues. Definition at line 69 of file LA_Eigenvalues.cpp.

const Vector & LA::Eigenvalues::getImagEigenvalues (   )  const

Returns a reference to the vector with the real eigenvalues. Returns: Reference to the vector with the real eigenvalues. Definition at line 74 of file LA_Eigenvalues.cpp. References e.

Matrix LA::Eigenvalues::getD (   )  const

Computes the block diagonal eigenvalue matrix. If the original matrix 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, a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex eigenvalues look like u + iv . . . . . . u - iv . . . . . . a + ib . . . . . . a - ib . . . . . . x . . . . . . y then D looks like u v . . . . -v u . . . . . . a b . . . . -b a . . . . . . x . . . . . . y This keeps V a real matrix in both symmetric and non-symmetric cases, and A*V = V*D. Returns: upon return, the matrix is filled with the block diagonal eigenvalue matrix. Definition at line 79 of file LA_Eigenvalues.cpp. References e.

void LA::Eigenvalues::tred2 (   )  [private]

Definition at line 101 of file LA_Eigenvalues.cpp. References e, e, V(), and V. Referenced by Eigenvalues().

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void LA::Eigenvalues::tql2 (   )  [private]

Definition at line 236 of file LA_Eigenvalues.cpp. References e, e, V(), and V. Referenced by Eigenvalues().

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void LA::Eigenvalues::orthes (   )  [private]

Definition at line 375 of file LA_Eigenvalues.cpp. References H, ort, and V. Referenced by Eigenvalues().

void LA::Eigenvalues::cdiv (  double  xr, double  xi, double  yr, double  yi )  [private]

Definition at line 482 of file LA_Eigenvalues.cpp. References cdivi, and cdivr. Referenced by hqr2().

void LA::Eigenvalues::hqr2 (   )  [private]

Definition at line 501 of file LA_Eigenvalues.cpp. References cdiv(), cdivi, cdivr, e, e, H, V(), and V. Referenced by Eigenvalues().

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Member Data Documentation

int LA::Eigenvalues::n [private]

Row and column dimension (square matrix). Definition at line 151 of file LA_Eigenvalues.h.

int LA::Eigenvalues::issymmetric [private]

Definition at line 153 of file LA_Eigenvalues.h. Referenced by Eigenvalues().

Vector LA::Eigenvalues::d [private]

Arrays for internal storage of eigenvalues. Definition at line 156 of file LA_Eigenvalues.h.

Vector LA::Eigenvalues::e [private]

Definition at line 157 of file LA_Eigenvalues.h. Referenced by Eigenvalues(), getD(), getImagEigenvalues(), hqr2(), tql2(), and tred2().

Matrix LA::Eigenvalues::V [private]

Array for internal storage of eigenvectors. Definition at line 160 of file LA_Eigenvalues.h. Referenced by Eigenvalues(), getV(), hqr2(), orthes(), tql2(), and tred2().

Matrix LA::Eigenvalues::H [private]

Array for internal storage of nonsymmetric Hessenberg form. internal storage of nonsymmetric Hessenberg form. Definition at line 165 of file LA_Eigenvalues.h. Referenced by Eigenvalues(), hqr2(), and orthes().

Vector LA::Eigenvalues::ort [private]

Working storage for nonsymmetric algorithm. working storage for nonsymmetric algorithm. Definition at line 170 of file LA_Eigenvalues.h. Referenced by Eigenvalues(), and orthes().

double LA::Eigenvalues::cdivr [private]

Definition at line 172 of file LA_Eigenvalues.h. Referenced by cdiv(), and hqr2().

double LA::Eigenvalues::cdivi [private]

Definition at line 172 of file LA_Eigenvalues.h. Referenced by cdiv(), and hqr2().

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The documentation for this class was generated from the following files:

• Tools/Math/LA_Eigenvalues.h
• Tools/Math/LA_Eigenvalues.cpp

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