Class SymmHess

java.lang.Object

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

org.flag4j.linalg.decompositions.unitary.UnitaryDecomposition<Matrix,double[]>

org.flag4j.linalg.decompositions.unitary.RealUnitaryDecomposition

org.flag4j.linalg.decompositions.hess.RealHess

org.flag4j.linalg.decompositions.hess.SymmHess

public class SymmHess
extends RealHess

Computes the Hessenberg decomposition of a real dense symmetric matrix.

The Hessenberg decomposition decomposes a given symmetric matrix \( A \) into the product: \[ A = QHQ^{T} \] where \( Q \) is an orthogonal matrix and \( H \) is a symmetric 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 SymmHess.
2. Call decompose(Matrix) to perform the factorization.
3. Retrieve the resulting matrices using getH() and RealUnitaryDecomposition.getQ().

See Also:

• HermHess
• getH()
• RealUnitaryDecomposition.getQ()

Field Summary

Fields

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

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

currentFactor, norm, shift

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
SymmHess()	Constructs a Hessenberg decomposer for symmetric matrices.
SymmHess(boolean computeQ)	Constructs a Hessenberg decomposer for symmetric matrices.
SymmHess(boolean computeQ, boolean enforceSymmetric)	Constructs a Hessenberg decomposer for symmetric matrices.
SymmHess(boolean computeQ, boolean enforceSymmetric, boolean inPlace)	Constructs a Hessenberg decomposer for symmetric matrices.

Method Summary

Modifier and Type	Method	Description
SymmHess	decompose(Matrix 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.
Matrix	getH()	Gets the Hessenberg matrix, \( H \), from the decomposition.
protected void	setUp(Matrix src)	Performs basic setup for the decomposition.
protected void	updateData(int j)	Updates the UnitaryDecomposition.transformMatrix matrix using the computed Householder vector from RealUnitaryDecomposition.computeHouseholder(int).

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

decompose, getUpper, initQ

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

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

enforceSymmetric

protected boolean enforceSymmetric

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

Constructor Details

SymmHess

public SymmHess()

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

SymmHess

public SymmHess(boolean computeQ)

Constructs a Hessenberg decomposer for symmetric matrices.

Parameters:

computeQ - Flag indicating if the orthogonal \( Q \) matrix from the Hessenberg decomposition should be explicitly computed. if true, then the \( Q \) matrix will be computed explicitly. If \( Q \) is not needed, setting this to false may yield an increase in performance.

SymmHess

public SymmHess(boolean computeQ,
 boolean enforceSymmetric)

Constructs a Hessenberg decomposer for symmetric matrices.

Parameters:

computeQ - Flag indicating if the orthogonal \( Q \) matrix from the Hessenberg decomposition should be explicitly computed. if true, then the \( Q \) matrix will be computed explicitly. If \( Q \) is not needed, setting this to false may yield an increase in performance.

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

SymmHess

public SymmHess(boolean computeQ,
 boolean enforceSymmetric,
 boolean inPlace)

Constructs a Hessenberg decomposer for symmetric matrices.

Parameters:

computeQ - Flag indicating if the orthogonal \( Q \) matrix from the Hessenberg decomposition should be explicitly computed. if true, then the \( Q \) matrix will be computed explicitly. If \( Q \) is not needed, setting this to false may yield an increase in performance.

enforceSymmetric - Flag indicating if an explicit check should be made to ensure any matrix passed to decompose(Matrix) is truly symmetric. If true, an exception will be thrown if the matrix is not symmetric. If false, the decomposition will proceed under the assumption that the matrix is symmetric whether it actually is or not. If the matrix is not symmetric, 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 SymmHess decompose(Matrix src)

Applies decomposition to the source matrix.

Overrides:

decompose in class RealHess

Parameters:

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

Returns:

A reference to this decomposer.

getH

public Matrix getH()

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

Overrides:

getH in class RealHess

Returns:

The symmetric 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 RealUnitaryDecomposition

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(Matrix src)

Performs basic setup for the decomposition.

Overrides:

setUp in class UnitaryDecomposition<Matrix,double[]>

Parameters:

src - The matrix to be decomposed.

Throws:

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

updateData

protected void updateData(int j)

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

Overrides:

updateData in class RealUnitaryDecomposition

Parameters:

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