Package org.flag4j.linalg.decompositions.unitary

Class RealUnitaryDecomposition

java.lang.Object

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

org.flag4j.linalg.decompositions.unitary.RealUnitaryDecomposition

All Implemented Interfaces:

Decomposition<Matrix>

Direct Known Subclasses:

RealHess, RealQR

public abstract class RealUnitaryDecomposition
extends UnitaryDecomposition<Matrix,double[]>

This class is the base class for real matrix decompositions which proceed by using unitary/orthogonal transformations (specifically Householder reflectors) to bring a matrix into an upper triangular/Hessenburg matrix. Specifically, the QR and Hessenburg decompositions.

Field Summary

Fields

Modifier and Type	Field	Description
protected double	currentFactor	Scalar factor of the currently computed Householder reflector.
protected double	norm	To store norms of columns in UnitaryDecomposition.transformMatrix.
protected double	shift	Stores the shifted value of the first entry in a Householder vector.

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

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

Constructor Summary

Constructors

Constructor	Description
RealUnitaryDecomposition(int subDiagonal)	Creates a real unitary decomposer which will reduce the matrix to an upper triangular/Hessenburg matrix which is has zeros below the specified sub-diagonal.
RealUnitaryDecomposition(int subDiagonal, boolean storeReflectors)	Creates a real unitary decomposer which will reduce the matrix to an upper triangular/Hessenburg matrix which has zeros below the specified sub-diagonal (must be 0 or 1).

Method Summary

Modifier and Type	Method	Description
protected void	computeHouseholder(int j)	Computes the Householder vector for the first column of the sub-matrix with upper left corner at (j, j).
protected void	computePhasedNorm(int j, double maxAbs)	Computes the norm of column j below the jth row of the matrix to be decomposed.
protected double	findMaxAndInit(int j)	Finds the maximum value in UnitaryDecomposition.transformMatrix at column j at or below the jth row.
Matrix	getQ()	Gets the unitary Q matrix from the unitary decomposition.
protected Matrix	getUpper(Matrix H)	Gets the upper triangular/Hessenburg matrix from the last decomposition.
protected void	initWorkArrays(int maxAxisSize)	Initialized any work arrays to be used in computing the decomposition with the proper size.
protected void	updateData(int j)	Updates the UnitaryDecomposition.transformMatrix matrix using the computed Householder vector from computeHouseholder(int).

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

decomposeBase, getUpper, initQ, setUp

Methods inherited from class java.lang.Object

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

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

decompose

Field Details

norm

protected double norm

To store norms of columns in UnitaryDecomposition.transformMatrix.

currentFactor

protected double currentFactor

Scalar factor of the currently computed Householder reflector.

shift

protected double shift

Stores the shifted value of the first entry in a Householder vector.

Constructor Details

RealUnitaryDecomposition

public RealUnitaryDecomposition(int subDiagonal)

Creates a real unitary decomposer which will reduce the matrix to an upper triangular/Hessenburg matrix which is has zeros below the specified sub-diagonal.

Parameters:

subDiagonal - Sub-diagonal of the upper triangular/Hessenburg matrix. That is, the sub-diagonal for which all entries below will be zero in the final upper quasi-triangular matrix. Must be Zero or one. If zero, it will be upper triangular. If one, it will be upper Hessenburg.

RealUnitaryDecomposition

public RealUnitaryDecomposition(int subDiagonal,
 boolean storeReflectors)

Creates a real unitary decomposer which will reduce the matrix to an upper triangular/Hessenburg matrix which has zeros below the specified sub-diagonal (must be 0 or 1).

Allows for specification if the reflectors used to bring matrix to upper triangular/Hessenburg form are to be stored or not.

If the Q matrix is need, then storeReflectors must be true. If Q is NOT needed, then not storing the reflectors may improve performance slightly by avoiding unneeded copies.

It should be noted that if performance is improved, it will be a very slight improvement compared to the total time to compute the decomposition. This is because the computation of Q is only evaluated lazily once getQ() is called, so this will only save on copy operations.

Parameters:

subDiagonal - Sub-diagonal of the upper triangular/Hessenburg matrix. That is, the sub-diagonal for which all entries below will be zero in the final upper quasi-triangular matrix. Must be Zero or one. If zero, it will be upper triangular. If one, it will be upper Hessenburg.

storeReflectors - Flag indicating if the reflectors used to bring the matrix to upper triangular/Hessenburg form should be stored.

Method Details

getQ

public Matrix getQ()

Gets the unitary Q matrix from the unitary decomposition.

Note, if the reflectors for this decomposition were not saved, then Q can not be computed and this method will be null.

Specified by:

getQ in class UnitaryDecomposition<Matrix,double[]>

Returns:

The Q matrix from the unitary decomposition. Note, if the reflectors for this decomposition were not saved, then Q can not be computed and this method will return null.

getUpper

protected Matrix getUpper(Matrix H)

Gets the upper triangular/Hessenburg matrix from the last decomposition.

Specified by:

getUpper in class UnitaryDecomposition<Matrix,double[]>

Parameters:

H - Storage for upper triangular/Hessenburg matrix. Assumed to be the zero matrix of an appropriate size.

Returns:

The upper triangular/Hessenburg matrix from the last decomposition.

initWorkArrays

protected void initWorkArrays(int maxAxisSize)

Initialized any work arrays to be used in computing the decomposition with the proper size.

Specified by:

initWorkArrays in class UnitaryDecomposition<Matrix,double[]>

Parameters:

maxAxisSize - Length of the largest axis in the matrix to be decomposed. That is, max(numRows, numCols)

computeHouseholder

protected void computeHouseholder(int j)

Computes the Householder vector for the first column of the sub-matrix with upper left corner at (j, j).

Specified by:

computeHouseholder in class UnitaryDecomposition<Matrix,double[]>

Parameters:

j - Index of the upper left corner of the sub-matrix for which to compute the Householder vector for the first column. That is, a Householder vector will be computed for the portion of column j below row j.

computePhasedNorm

protected void computePhasedNorm(int j,
 double maxAbs)

Computes the norm of column j below the jth row of the matrix to be decomposed. The norm will have the same parity as the first entry in the sub-column.

Specified by:

computePhasedNorm in class UnitaryDecomposition<Matrix,double[]>

Parameters:

j - Column to compute norm of below the jth row.

maxAbs - Maximum absolute value in the column. Used for scaling norm to minimize potential overflow issues.

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 entries of the storage array UnitaryDecomposition.householderVector to the entries of this column.

Specified by:

findMaxAndInit in class UnitaryDecomposition<Matrix,double[]>

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.

updateData

protected void updateData(int j)

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

Specified by:

updateData in class UnitaryDecomposition<Matrix,double[]>

Parameters:

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