Class HermHess

java.lang.Object

org.flag4j.linalg.decompositions.Decomposition<CMatrix>

org.flag4j.linalg.decompositions.unitary.UnitaryDecomposition<CMatrix,Complex128[]>

org.flag4j.linalg.decompositions.unitary.ComplexUnitaryDecomposition

org.flag4j.linalg.decompositions.hess.ComplexHess

org.flag4j.linalg.decompositions.hess.HermHess

public class HermHess
extends ComplexHess

Computes the Hessenberg decomposition of a real dense Hermitian matrix.

The Hessenberg decomposition decomposes a given Hermitian matrix \( A \) into the product: \[ A = QHQ^{H} \] where \( Q \) is an orthogonal matrix and \( H \) is a Hermitian tri-diagonal matrix (special case of Hessenburg form) which is similar to \( A \) (i.e. has the same eigenvalues)

A matrix \( H \) is in tri-diagonal form if it has all zeros below the first sub-diagonal and above the first super-diagonal.

For example, the following matrix is in symmetric tri-diagonal form where each '\( \times \)' may hold a different value (provided the matrix is symmetric): \[ \begin{bmatrix} \times & \times & 0 & 0 & 0 \\ \times & \times & \times & 0 & 0 \\ 0 & \times & \times & \times & 0 \\ 0 & 0 & \times & \times & \times \\ 0 & 0 & 0 & \times & \times \\ \end{bmatrix} \]

Efficiency Considerations:

• If the orthogonal matrix \( Q \) is not required, setting computeQ = false in the constructor may improve performance.
• Support for in-place decomposition to reduce memory usage.

Usage:

The decomposition workflow typically follows these steps:

1. Instantiate an instance of HermHess.
2. Call decompose(CMatrix) to perform the factorization.
3. Retrieve the resulting matrices using getH() and ComplexUnitaryDecomposition.getQ().

See Also:

• SymmHess
• getH()
• ComplexUnitaryDecomposition.getQ()

Field Summary

Fields

Modifier and Type	Field	Description
protected boolean	enforceHermitian	Flag indicating if an explicit check should be made that the matrix to be decomposed is Hermitian.

Fields inherited from class org.flag4j.linalg.decompositions.unitary.ComplexUnitaryDecomposition

currentFactor, norm, normRe

Fields inherited from class org.flag4j.linalg.decompositions.unitary.UnitaryDecomposition

applyUpdate, householderVector, iHigh, iLow, inPlace, minAxisSize, numCols, numRows, qFactors, storeReflectors, subDiagonal, transformData, transformMatrix, workArray

Fields inherited from class org.flag4j.linalg.decompositions.Decomposition

hasDecomposed

Constructor Summary

Constructors

Constructor	Description
HermHess()	Constructs a Hessenberg decomposer for Hermitian matrices.
HermHess(boolean computeQ)	Constructs a Hessenberg decomposer for Hermitian matrices.
HermHess(boolean computeQ, boolean enforceHermitian)	Constructs a Hessenberg decomposer for Hermitian matrices.
HermHess(boolean computeQ, boolean enforceHermitian, boolean inPlace)	Constructs a Hessenberg decomposer for Hermitian matrices.

Method Summary

Modifier and Type	Method	Description
HermHess	decompose(CMatrix src)	Applies decomposition to the source matrix.
protected double	findMaxAndInit(int j)	Finds the maximum value in UnitaryDecomposition.transformMatrix at column j at or below the jth row.
CMatrix	getH()	Gets the Hessenberg matrix, \( H \), from the decomposition.
protected void	setUp(CMatrix src)	Performs basic setup for the decomposition.
protected void	updateData(int j)	Updates the UnitaryDecomposition.transformMatrix matrix using the computed Householder vector from ComplexUnitaryDecomposition.computeHouseholder(int).

Methods inherited from class org.flag4j.linalg.decompositions.hess.ComplexHess

decompose, getUpper, initQ

Methods inherited from class org.flag4j.linalg.decompositions.unitary.ComplexUnitaryDecomposition

computeHouseholder, computePhasedNorm, getQ, getUpper, initWorkArrays

Methods inherited from class org.flag4j.linalg.decompositions.Decomposition

ensureHasDecomposed

Methods inherited from class java.lang.Object

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

Field Details

enforceHermitian

protected boolean enforceHermitian

Flag indicating if an explicit check should be made that the matrix to be decomposed is Hermitian.

Constructor Details

HermHess

public HermHess()

Constructs a Hessenberg decomposer for Hermitian matrices. By default, the Householder vectors used in the decomposition will be stored so that the full unitary \( Q \) matrix can be formed by calling ComplexUnitaryDecomposition.getQ().

HermHess

public HermHess(boolean computeQ)

Constructs a Hessenberg decomposer for Hermitian matrices.

Parameters:

computeQ - Flag indicating if the unitary \( Q \) matrix from the Hessenberg decomposition should be explicitly computed. if true, then the \( Q \) matrix will be computed explicitly.

HermHess

public HermHess(boolean computeQ,
 boolean enforceHermitian)

Constructs a Hessenberg decomposer for Hermitian matrices.

Parameters:

computeQ - Flag indicating if the unitary \( Q \) matrix from the Hessenberg decomposition should be explicitly computed. if true, then the \( Q \) matrix will be computed explicitly.

enforceHermitian - Flag indicating if an explicit check should be made to ensure any matrix passed to decompose(CMatrix) is truly Hermitian. if true, an exception will be thrown if the matrix is not Hermitian. If false, the decomposition will proceed under the assumption that the matrix is Hermitian whether it actually is or not. If the matrix is not Hermitian, then the values in the upper triangular portion of the matrix are taken to be the values.

HermHess

public HermHess(boolean computeQ,
 boolean enforceHermitian,
 boolean inPlace)

Constructs a Hessenberg decomposer for Hermitian matrices.

Parameters:

computeQ - Flag indicating if the unitary \( Q \) matrix from the Hessenberg decomposition should be explicitly computed. if true, then the \( Q \) matrix will be computed explicitly.

enforceHermitian - Flag indicating if an explicit check should be made to ensure any matrix passed to decompose(CMatrix) is truly Hermitian. if true, an exception will be thrown if the matrix is not Hermitian. If false, the decomposition will proceed under the assumption that the matrix is Hermitian whether it actually is or not. If the matrix is not Hermitian, then the values in the upper triangular portion of the matrix are taken to be the values.

inPlace - Flag indicating if the decomposition should be done in-place.

• If true, then the decomposition will be done in place.
• If false, then the decomposition will be done out-of-place.

Method Details

decompose

public HermHess decompose(CMatrix src)

Applies decomposition to the source matrix.

Overrides:

decompose in class ComplexHess

Parameters:

src - The source matrix to decompose. (Assumed to be a Hermitian matrix).

Returns:

A reference to this decomposer.

getH

public CMatrix getH()

Gets the Hessenberg matrix, \( H \), from the decomposition. The matrix will be Hermitian tri-diagonal.

Overrides:

getH in class ComplexHess

Returns:

The Hermitian tri-diagonal (Hessenberg) matrix, \( H \), from this decomposition.

findMaxAndInit

protected double findMaxAndInit(int j)

Finds the maximum value in UnitaryDecomposition.transformMatrix at column j at or below the jth row. This method also initializes the first numRows-j data of the storage array UnitaryDecomposition.householderVector to the data of this column.

Overrides:

findMaxAndInit in class ComplexUnitaryDecomposition

Parameters:

j - Index of column (and starting row) to compute max of.

Returns:

The maximum value in UnitaryDecomposition.transformMatrix at column j at or below the jth row.

setUp

protected void setUp(CMatrix src)

Performs basic setup for the decomposition.

Overrides:

setUp in class UnitaryDecomposition<CMatrix,Complex128[]>

Parameters:

src - The matrix to be decomposed.

Throws:

LinearAlgebraException - If the matrix is not Hermitian and enforceHermitian is true or if the matrix is not square regardless of the value of enforceHermitian.

updateData

protected void updateData(int j)

Updates the UnitaryDecomposition.transformMatrix matrix using the computed Householder vector from ComplexUnitaryDecomposition.computeHouseholder(int).

Overrides:

updateData in class ComplexUnitaryDecomposition

Parameters:

j - Index of sub-matrix for which the Householder reflector was computed for.