Package org.flag4j.linalg

Class Eigen

java.lang.Object

org.flag4j.linalg.Eigen

public class Eigen
extends Object

This class provides several methods useful for computing eigenvalues and eigenvectors.

Constructor Summary

Constructors

Constructor	Description
Eigen()

Method Summary

Modifier and Type	Method	Description
static CNumber[]	get2x2EigenValues(double a11, double a12, double a21, double a22)	Computes the eigenvalues of a 2x2 matrix explicitly.
static CVector	get2x2EigenValues(CMatrix src)	Computes the eigenvalues of a 2x2 matrix explicitly.
static CVector	get2x2EigenValues(Matrix src)	Computes the eigenvalues of a 2x2 matrix explicitly.
static CNumber[]	get2x2EigenValues(CNumber a11, CNumber a12, CNumber a21, CNumber a22)
static CMatrix[]	getEigenPairs(CMatrix src)	Computes the eigenvalues and eigenvectors of a square real matrix.
static CMatrix[]	getEigenPairs(Matrix src)	Computes the eigenvalues and eigenvectors of a square real matrix.
static CVector	getEigenValues(CMatrix src)	Computes the eigenvalues of a square complex dense matrix.
static CVector	getEigenValues(CMatrix src, long seed)	Computes the eigenvalues of a square complex dense matrix.
static CVector	getEigenValues(CMatrix src, long seed, int maxIterationFactor)	Computes the eigenvalues of a square complex dense matrix.
private static CVector	getEigenValues(CMatrix src, ComplexSchur schur)	Computes the eigenvalues of a square complex dense matrix.
static CVector	getEigenValues(Matrix src)	Computes the eigenvalues of a square real dense matrix.
static CVector	getEigenValues(Matrix src, long seed)	Computes the eigenvalues of a square real dense matrix.
static CVector	getEigenValues(Matrix src, long seed, int maxIterationFactor)	Computes the eigenvalues of a square real dense matrix.
private static CVector	getEigenValues(Matrix src, RealSchur schur)	Computes the eigenvalues of a real dense square matrix.
static CMatrix	getEigenVectors(CMatrix src)	Computes the eigenvectors of a square real dense matrix.
static CMatrix	getEigenVectors(CMatrix src, long seed)	Computes the eigenvectors of a square real dense matrix.
static CMatrix	getEigenVectors(CMatrix src, long seed, int maxIterationFactor)	Computes the eigenvectors of a square real dense matrix.
private static CMatrix	getEigenVectors(CMatrix src, ComplexSchur schur)	Computes the eigenvectors of a square complex dense matrix.
static CMatrix	getEigenVectors(Matrix src)	Computes the eigenvectors of a square real dense matrix.
static CMatrix	getEigenVectors(Matrix src, long seed)	Computes the eigenvectors of a square real dense matrix.
static CMatrix	getEigenVectors(Matrix src, long seed, int maxIterationFactor)	Computes the eigenvectors of a square real dense matrix.
private static CMatrix	getEigenVectors(Matrix src, RealSchur schur)	Computes the eigenvectors of a square real dense matrix.
static CMatrix	getEigenVectorsTriu(CMatrix T)	Computes the eigenvectors of an upper triangular matrix.
static Matrix	getEigenVectorsTriu(Matrix T)	Computes the eigenvectors of an upper triangular matrix.
private static void	makeSystem(CMatrix T, int j, CMatrix S_hat, CVector r)	Constructs the matrix (T - lambda*I)[0:j][0:j] for use in computing the eigenvalue of the upper triangular matrix T associated with the jth eigenvalue lambda.
private static void	makeSystem(Matrix T, int j, Matrix S_hat, Vector r)	Constructs the matrix (T - lambda*I)[0:j][0:j] for use in computing the eigenvalue of the upper triangular matrix T associated with the jth eigenvalue lambda.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

Eigen

public Eigen()

Method Details

get2x2EigenValues

public static CVector get2x2EigenValues(Matrix src)

Computes the eigenvalues of a 2x2 matrix explicitly.

Parameters:

src - The 2x2 matrix to compute the eigenvalues of.

Returns:

A complex vector containing the eigenvalues of the 2x2 src matrix.

get2x2EigenValues

public static CNumber[] get2x2EigenValues(double a11,
 double a12,
 double a21,
 double a22)

Computes the eigenvalues of a 2x2 matrix explicitly.

Parameters:

a11 - First entry in matrix (at index (0, 0)).

a12 - Second entry in matrix (at index (0, 1)).

a21 - Third entry in matrix (at index (1, 0)).

a22 - Fourth entry in matrix (at index (1, 1)).

Returns:

A complex array containing the eigenvalues of the 2x2 src matrix.

get2x2EigenValues

public static CVector get2x2EigenValues(CMatrix src)

Computes the eigenvalues of a 2x2 matrix explicitly.

Parameters:

src - The 2x2 matrix to compute the eigenvalues of.

Returns:

A complex vector containing the eigenvalues of the 2x2 src matrix.

get2x2EigenValues

public static CNumber[] get2x2EigenValues(CNumber a11,
 CNumber a12,
 CNumber a21,
 CNumber a22)

getEigenValues

public static CVector getEigenValues(Matrix src)

Computes the eigenvalues of a square real dense matrix. For reproducibility see getEigenValues(Matrix, long). If the algorithm fails to converge within the default maximum number of iterations, getEigenValues(Matrix, long, int) can be used to specify a larger number of iterations to attempt.

Parameters:

src - The matrix to compute the eigenvalues of.

Returns:

The eigenvalues of the src matrix.

See Also:

• getEigenValues(Matrix, long)
• getEigenValues(Matrix, long, int)

getEigenValues

public static CVector getEigenValues(Matrix src,
 long seed)

Computes the eigenvalues of a square real dense matrix. If the algorithm fails to converge within the default maximum number of iterations, getEigenValues(Matrix, long, int) can be used to specify a larger number of iterations to attempt.

Parameters:

src - The matrix to compute the eigenvalues of.

seed - Seed for random shifts used in computing the eigenvalues.

Returns:

The eigenvalues of the src matrix.

See Also:

• getEigenValues(Matrix)
• getEigenValues(Matrix, long, int)

getEigenValues

public static CVector getEigenValues(Matrix src,
 long seed,
 int maxIterationFactor)

Computes the eigenvalues of a square real dense matrix.

Parameters:

src - The matrix to compute the eigenvalues of.

seed - Seed for random shifts used in computing the eigenvalues.

maxIterationFactor - maximum iteration factor for use in computing the total maximum number of iterations to run the QR algorithm for during eigenvalue computation. The maximum number of iterations will be computed as:

      maxIteration = maxIterationFactor * src.numRows;

Returns:

The eigenvalues of the src matrix.

See Also:

• getEigenValues(Matrix)
• getEigenValues(Matrix, long)

getEigenValues

private static CVector getEigenValues(Matrix src,
 RealSchur schur)

Computes the eigenvalues of a real dense square matrix.

Parameters:

src - Matrix to compute eigenvalues of.

schur - Schur decomposer to use in the eigenvalue computation.

Returns:

The eigenvalues of the src matrix stored in a complex vector (CVector).

getEigenValues

public static CVector getEigenValues(CMatrix src)

Computes the eigenvalues of a square complex dense matrix. For reproducibility see getEigenValues(CMatrix, long).

Parameters:

src - The matrix to compute the eigenvalues of.

Returns:

The eigenvalues of the src matrix.

See Also:

• getEigenValues(CMatrix, long)

getEigenValues

public static CVector getEigenValues(CMatrix src,
 long seed)

Computes the eigenvalues of a square complex dense matrix.

Parameters:

src - The matrix to compute the eigenvalues of.

seed - Seed for random shifts used in computing the eigenvalues. This allows for reproducibility despite randomness in the eigenvalue computation algorithm.

Returns:

The eigenvalues of the src matrix.

See Also:

• getEigenValues(CMatrix)

getEigenValues

public static CVector getEigenValues(CMatrix src,
 long seed,
 int maxIterationFactor)

Computes the eigenvalues of a square complex dense matrix.

Parameters:

src - The matrix to compute the eigenvalues of.

seed - Seed for random shifts used in computing the eigenvalues.

maxIterationFactor - maximum iteration factor for use in computing the total maximum number of iterations to run the QR algorithm for during eigenvalue computation. The maximum number of iterations will be computed as:

      maxIteration = maxIterationFactor * src.numRows;

Returns:

The eigenvalues of the src matrix.

See Also:

• getEigenValues(Matrix)
• getEigenValues(Matrix, long)

getEigenValues

private static CVector getEigenValues(CMatrix src,
 ComplexSchur schur)

Computes the eigenvalues of a square complex dense matrix.

Parameters:

src - The matrix to compute the eigenvalues of.

schur - The schur decomposer to use in the eigenvalue decomposition.

Returns:

The eigenvalues of the src matrix.

getEigenVectors

public static CMatrix getEigenVectors(Matrix src)

Computes the eigenvectors of a square real dense matrix.

Parameters:

src - The matrix to compute the eigenvectors of.

Returns:

A matrix containing the eigenvectors of src as its columns.

See Also:

• getEigenVectors(Matrix, long)
• getEigenVectors(Matrix, long, int)
• getEigenValues(Matrix)
• getEigenValues(Matrix, long)
• getEigenValues(Matrix, long, int)

getEigenVectors

public static CMatrix getEigenVectors(Matrix src,
 long seed)

Computes the eigenvectors of a square real dense matrix. This method accepts a seed for the random number generator involved in the algorithm for computing the eigenvalues. This allows for reproducibility between calls.

Parameters:

src - The matrix to compute the eigenvectors of.

seed - Seed for random shifts used in the algorithm to compute the eigenvalues of.

Returns:

A matrix containing the eigenvectors of src as its columns.

See Also:

• getEigenVectors(Matrix)
• getEigenVectors(Matrix, long, int)
• getEigenValues(Matrix)
• getEigenValues(Matrix, long)
• getEigenValues(Matrix, long, int)

getEigenVectors

public static CMatrix getEigenVectors(Matrix src,
 long seed,
 int maxIterationFactor)

Computes the eigenvectors of a square real dense matrix. This method accepts a seed for the random number generator involved in the algorithm for computing the eigenvalues. This allows for reproducibility between calls.

Parameters:

src - The matrix to compute the eigenvectors of.

seed - Seed for random shifts used in the algorithm to compute the eigenvalues of.

maxIterationFactor - maximum iteration factor for use in computing the total maximum number of iterations to run the QR algorithm for during eigenvalue computation. The maximum number of iterations will be computed as:

      maxIteration = maxIterationFactor * src.numRows;

Returns:

A matrix containing the eigenvectors of src as its columns.

See Also:

• getEigenVectors(CMatrix)
• getEigenVectors(CMatrix, long)
• getEigenValues(CMatrix)
• getEigenValues(CMatrix, long)
• getEigenValues(CMatrix, long, int)

getEigenVectors

private static CMatrix getEigenVectors(Matrix src,
 RealSchur schur)

Computes the eigenvectors of a square real dense matrix.

Parameters:

src - The matrix to compute the eigenvectors of.

schur - Schur decomposer to use in the computation of the eigenvectors.

Returns:

A matrix containing the eigenvectors of src as its columns.

getEigenVectors

public static CMatrix getEigenVectors(CMatrix src)

Computes the eigenvectors of a square real dense matrix.

Parameters:

src - The matrix to compute the eigenvectors of.

Returns:

A matrix containing the eigenvectors of src as its columns.

See Also:

• getEigenVectors(CMatrix, long)
• getEigenVectors(CMatrix, long, int)
• getEigenValues(CMatrix)
• getEigenValues(CMatrix, long)
• getEigenValues(CMatrix, long, int)

getEigenVectors

public static CMatrix getEigenVectors(CMatrix src,
 long seed)

Computes the eigenvectors of a square real dense matrix. This method accepts a seed for the random number generator involved in the algorithm for computing the eigenvalues. This allows for reproducibility between calls.

Parameters:

src - The matrix to compute the eigenvectors of.

seed - Seed for random shifts used in the algorithm to compute the eigenvalues of.

Returns:

A matrix containing the eigenvectors of src as its columns.

See Also:

• getEigenVectors(CMatrix)
• getEigenVectors(CMatrix, long, int)
• getEigenValues(CMatrix)
• getEigenValues(CMatrix, long)
• getEigenValues(CMatrix, long, int)

getEigenVectors

public static CMatrix getEigenVectors(CMatrix src,
 long seed,
 int maxIterationFactor)

Computes the eigenvectors of a square real dense matrix. This method accepts a seed for the random number generator involved in the algorithm for computing the eigenvalues. This allows for reproducibility between calls.

Parameters:

src - The matrix to compute the eigenvectors of.

seed - Seed for random shifts used in the algorithm to compute the eigenvalues of.

maxIterationFactor - maximum iteration factor for use in computing the total maximum number of iterations to run the QR algorithm for during eigenvalue computation. The maximum number of iterations will be computed as:

      maxIteration = maxIterationFactor * src.numRows;

Returns:

A matrix containing the eigenvectors of src as its columns.

See Also:

• getEigenVectors(CMatrix)
• getEigenVectors(CMatrix, long)
• getEigenValues(CMatrix)
• getEigenValues(CMatrix, long)
• getEigenValues(CMatrix, long, int)

getEigenVectors

private static CMatrix getEigenVectors(CMatrix src,
 ComplexSchur schur)

Computes the eigenvectors of a square complex dense matrix.

Parameters:

src - The matrix to compute the eigenvectors of.

schur - Schur decomposer to use when computing the eigenvectors.

Returns:

A matrix containing the eigenvectors of src as its columns.

getEigenVectorsTriu

public static Matrix getEigenVectorsTriu(Matrix T)

Computes the eigenvectors of an upper triangular matrix.

Parameters:

T - The upper triangular matrix to compute the eigenvectors of.

Returns:

A matrix containing the eigenvectors of T as its columns.

getEigenVectorsTriu

public static CMatrix getEigenVectorsTriu(CMatrix T)

Computes the eigenvectors of an upper triangular matrix.

Parameters:

T - The upper triangular matrix to compute the eigenvectors of.

Returns:

A matrix containing the eigenvectors of T as its columns.

makeSystem

private static void makeSystem(CMatrix T,
 int j,
 CMatrix S_hat,
 CVector r)

Constructs the matrix (T - lambda*I)[0:j][0:j] for use in computing the eigenvalue of the upper triangular matrix T associated with the j^th eigenvalue lambda.

Parameters:

T - The upper triangular matrix (Assumed to be square).

j - The diagonal index of T where the eigenvalue lambda appears.

makeSystem

private static void makeSystem(Matrix T,
 int j,
 Matrix S_hat,
 Vector r)

Constructs the matrix (T - lambda*I)[0:j][0:j] for use in computing the eigenvalue of the upper triangular matrix T associated with the j^th eigenvalue lambda.

Parameters:

T - The upper triangular matrix (Assumed to be square).

j - The diagonal index of T where the eigenvalue lambda appears.

getEigenPairs

public static CMatrix[] getEigenPairs(Matrix src)

Computes the eigenvalues and eigenvectors of a square real matrix.

Parameters:

src - The matrix to compute the eigenvalues and vectors of.

Returns:

An array containing two matrices. The first matrix has shape 1xsrc.numCols and contains the eigenvalues of the src matrix. The second matrix has shape src.numRowsxsrc.numCols and contains the eigenvectors of the src matrix as its columns.

getEigenPairs

public static CMatrix[] getEigenPairs(CMatrix src)

Computes the eigenvalues and eigenvectors of a square real matrix.

Parameters:

src - The matrix to compute the eigenvalues and vectors of.

Returns:

An array containing two matrices. The first matrix has shape 1xsrc.numCols and contains the eigenvalues of the src matrix. The second matrix has shape src.numRowsxsrc.numCols and contains the eigenvectors of the src matrix as its columns.