Package org.flag4j.linalg.decompositions.schur

Class RealSchur

java.lang.Object

org.flag4j.linalg.decompositions.schur.Schur<Matrix,double[]>

org.flag4j.linalg.decompositions.schur.RealSchur

All Implemented Interfaces:

Decomposition<Matrix>

public class RealSchur
extends Schur<Matrix,double[]>

This class computes the Schur decomposition of a real dense square matrix.

That is, decompose a square matrix A into A=UTU^T where U is an orthogonal matrix and T is a block-upper triangular matrix called the real-Schur form of A. T is upper triangular except for possibly 2x2 blocks along the diagonal. T is similar to A.

This code was adapted from the code found in the EJML library and the description of the Francis implicit double shifted QR algorithm given in Fundamentals of Matrix Computations 3rd Edition by David S. Watkins.

Field Summary

Fields

Modifier and Type	Field	Description
protected double	currentFactor	Stores the scalar factor invalid input: '&alpha' for use in computation of the Householder reflector P = I - invalid input: '&alpha'vvT .
protected double	norm	For computing the norm of a column for use when computing Householder reflectors.

Fields inherited from class org.flag4j.linalg.decompositions.schur.Schur

checkFinite, computeU, DEFAULT_EXCEPTIONAL_ITERS, DEFAULT_MAX_ITERS_FACTOR, exceptionalThreshold, hess, householderVector, maxIterations, maxIterationsFactor, numExceptional, numRows, rng, shiftCol, sinceLastExceptional, T, temp, U, workArray

Constructor Summary

Constructors

Constructor	Description
RealSchur()	Creates a decomposer to compute the Schur decomposition for a real dense matrix.
RealSchur(boolean computeU)	Creates a decomposer to compute the Schur decomposition for a real dense matrix where the U matrix may or may not be computed.
RealSchur(boolean computeU, long seed)	Creates a decomposer to compute the Schur decomposition for a real dense matrix.
RealSchur(long seed)	Creates a decomposer to compute the Schur decomposition for a real dense matrix.

Method Summary

Modifier and Type	Method	Description
protected void	applyDoubleShiftReflector(int i, boolean set)	Applies reflector for the double shift.
protected void	applyReflector(int i, int shiftSize)	Applies the constructed Householder reflector which has been constructed for the given shift size.
protected void	applySingleShiftReflector(int i, boolean set)	Applies reflector for the double shift.
protected int	checkConvergence(int workingSize)	Checks for convergence of lower 2x2 sub-matrix within working matrix to upper triangular or block upper triangular form.
protected double	computeExceptionalShift(int k)	Computes a random shift to help the QR algorithm converge if it gets stuck.
protected void	computeImplicitDoubleShift(int workingSize)	Computes the shifts for a Francis double shift iteration.
protected void	computeImplicitSingleShift(int k, double shift)	Computes the non-zero entries of the first column for the single shifted QR algorithm.
RealSchur	decompose(Matrix src)	Computes the Schur decomposition of the input matrix.
protected boolean	makeReflector(int i, double p1, double p2)	Constructs a householder reflector given specified values for a column to apply the reflector to.
protected boolean	makeReflector(int i, double p1, double p2, double p3)	Constructs a householder reflector given specified values for a column to apply the reflector to.
protected void	performDoubleShift(int workingSize)	Performs a full iteration of the Francis implicit double shifted QR algorithm (this includes the bulge chase).
protected void	performExceptionalShift(int workingSize)	Performs a full iteration of the single shifted QR algorithm (this includes the bulge chase) where the shift is chosen to be a random value with the same magnitude as the lower right element of the working matrix.
protected void	performSingleShift(int workingSize, double shift)	Performs a full iteration of the implicit single shifted QR algorithm (this includes the bulge chase).
CMatrix[]	real2ComplexSchur()	Converts the real schur form computed in the last decomposition to the complex Schur form.
RealSchur	setExceptionalThreshold(int exceptionalThreshold)	Sets the number of iterations of the QR algorithm to perform without deflation before performing a random shift.
RealSchur	setMaxIterationFactor(int maxIterationFactor)	Specify maximum iteration factor for computing the total number of iterations to run the QR algorithm for when computing the decomposition.
protected void	setUpArrays()	Initializes temporary work arrays to be used in the decomposition.

Methods inherited from class org.flag4j.linalg.decompositions.schur.Schur

decomposeBase, getT, getU, setUp

Methods inherited from class java.lang.Object

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

Field Details

norm

protected double norm

For computing the norm of a column for use when computing Householder reflectors.

currentFactor

protected double currentFactor

Stores the scalar factor invalid input: '&alpha' for use in computation of the Householder reflector P = I - invalid input: '&alpha'vv^T .

Constructor Details

RealSchur

public RealSchur()

Creates a decomposer to compute the Schur decomposition for a real dense matrix.

Note: This decomposer may use random numbers during the decomposition. If reproducible results are needed, set the seed for the pseudo-random number generator using RealSchur(long)

RealSchur

public RealSchur(boolean computeU)

Creates a decomposer to compute the Schur decomposition for a real dense matrix where the U matrix may or may not be computed.

If the U matrix is not needed, passing computeU = false may provide a performance improvement.

By default if a constructor with no computeU parameter is called, U WILL be computed.

Note: This decomposer may use random numbers during the decomposition. If reproducible results are needed, set the seed for the pseudo-random number generator using RealSchur(boolean, long)

Parameters:

computeU - Flag indicating if the unitary U matrix should be computed for the Schur decomposition. If true, U will be computed. If false, U will not be computed.

RealSchur

public RealSchur(long seed)

Creates a decomposer to compute the Schur decomposition for a real dense matrix.

Parameters:

seed - Seed to use for pseudo-random number generator when computing exceptional shifts during the QR algorithm.

RealSchur

public RealSchur(boolean computeU,
 long seed)

Creates a decomposer to compute the Schur decomposition for a real dense matrix.

Parameters:

seed - Seed to use for pseudo-random number generator when computing exceptional shifts during the QR algorithm.

Method Details

setExceptionalThreshold

public RealSchur setExceptionalThreshold(int exceptionalThreshold)

Sets the number of iterations of the QR algorithm to perform without deflation before performing a random shift.

That is, if exceptionalThreshold = 10, then at most 10 iterations QR algorithm iterations will be performed. If, by the 10th iteration, no convergence has been detected which allows for deflation, then a QR algorithm iteration will be performed with a random (i.e. exceptional) shift.

By default, the threshold is set to Schur.DEFAULT_EXCEPTIONAL_ITERS

Overrides:

setExceptionalThreshold in class Schur<Matrix,double[]>

Parameters:

exceptionalThreshold - The new exceptional shift threshold. i.e. the number of iterations to perform without deflation before performing an iteration with random shifts.

Returns:

A reference to this decomposer.

Throws:

IllegalArgumentException - If exceptionalThreshold is not positive.

setMaxIterationFactor

public RealSchur setMaxIterationFactor(int maxIterationFactor)

Specify maximum iteration factor for computing the total number of iterations to run the QR algorithm for when computing the decomposition. The maximum number of iterations is computed as

     maxIteration = maxIterationFactor * src.numRows; 

If the algorithm does not converge within this limit, an error will be thrown.

By default, this is computed as

     maxIterations = Schur.DEFAULT_MAX_ITERS_FACTOR* src.numRows;

where src is the matrix being decomposed.

Overrides:

setMaxIterationFactor in class Schur<Matrix,double[]>

Parameters:

maxIterationFactor - maximum iteration factor for use in computing the total maximum number of iterations to run the QR algorithm for.

Returns:

A reference to this decomposer.

Throws:

IllegalArgumentException - If maxIterationFactor is not positive.

decompose

public RealSchur decompose(Matrix src)

Computes the Schur decomposition of the input matrix.

Parameters:

src - The source matrix to decompose.

Returns:

A reference to this decomposer.

setUpArrays

protected void setUpArrays()

Initializes temporary work arrays to be used in the decomposition.

Specified by:

setUpArrays in class Schur<Matrix,double[]>

performExceptionalShift

protected void performExceptionalShift(int workingSize)

Performs a full iteration of the single shifted QR algorithm (this includes the bulge chase) where the shift is chosen to be a random value with the same magnitude as the lower right element of the working matrix. This can help the QR converge for certain pathological cases where the double shift algorithm oscillates or fails to converge for repeated eigenvalues.

Specified by:

performExceptionalShift in class Schur<Matrix,double[]>

Parameters:

workingSize - The current working size for the decomposition. I.e. all entries below this row have converged to an upper or possible 2x2 block upper triangular form.

computeExceptionalShift

protected double computeExceptionalShift(int k)

Computes a random shift to help the QR algorithm converge if it gets stuck.

Parameters:

k - The current size of the working matrix. Specifically, the index of the lower right value in the working matrix is (k, k).

Returns:

A shift in a random direction which has the same magnitude as the elements in the matrix.

computeImplicitSingleShift

protected void computeImplicitSingleShift(int k,
 double shift)

Computes the non-zero entries of the first column for the single shifted QR algorithm.

Parameters:

k - Size of current working matrix.

shift - The shift to use.

performSingleShift

protected void performSingleShift(int workingSize,
 double shift)

Performs a full iteration of the implicit single shifted QR algorithm (this includes the bulge chase).

Parameters:

workingSize - The current working size for the decomposition. I.e. all entries below this row have converged to an upper or possible 2x2 block upper triangular form.

shift - The shift to use in the implicit single shifted QR algorithm.

applySingleShiftReflector

protected void applySingleShiftReflector(int i,
 boolean set)

Applies reflector for the double shift. This method can be used to apply either be the reflector constructed for the first column of the shifted matrix, or a reflector being used in the bulge chase of size 2 which arises from the first case.

Parameters:

i - The starting row the reflector is being applied to.

performDoubleShift

protected void performDoubleShift(int workingSize)

Performs a full iteration of the Francis implicit double shifted QR algorithm (this includes the bulge chase).

Specified by:

performDoubleShift in class Schur<Matrix,double[]>

Parameters:

workingSize - The current working size for the decomposition. I.e. all entries below this row have converged to an upper or possible 2x2 block upper triangular form.

computeImplicitDoubleShift

protected void computeImplicitDoubleShift(int workingSize)

Computes the shifts for a Francis double shift iteration. Specifically, the shifts are the generalized Rayleigh quotients of degree two.

Parameters:

workingSize - Size of current working matrix.

applyDoubleShiftReflector

protected void applyDoubleShiftReflector(int i,
 boolean set)

Applies reflector for the double shift. This method can be used to apply either be the reflector constructed for the first column of the shifted matrix, or a reflector being used in the bulge chase of size 2 which arises from the first case.

Parameters:

i - The starting row the reflector is being applied to.

applyReflector

protected void applyReflector(int i,
 int shiftSize)

Applies the constructed Householder reflector which has been constructed for the given shift size.

Parameters:

i - The stating row the reflector is being applied to.

shiftSize - The size of the shift the reflector was constructed for.

makeReflector

protected boolean makeReflector(int i,
 double p1,
 double p2,
 double p3)

Constructs a householder reflector given specified values for a column to apply the reflector to. This reflector is stored in indices i, i+1, and i+2 of Schur.householderVector.

Parameters:

i - Row of working matrix to construct reflector for.

p1 - First entry to in column to apply reflector to.

p2 - Second entry in column to apply reflector to.

p3 - Third entry in column to apply reflector to.

Returns:

True if a reflector needs to be constructed to return matrix to upper Hessenburg form. False if column is already in the correct form.

makeReflector

protected boolean makeReflector(int i,
 double p1,
 double p2)

Constructs a householder reflector given specified values for a column to apply the reflector to. This reflector is stored in indices i and i+1 of Schur.householderVector.

Parameters:

i - Row of working matrix to construct reflector for.

p1 - First entry to in column to apply reflector to.

p2 - Second entry in column to apply reflector to.

Returns:

True if a reflector needs to be constructed to return matrix to upper Hessenburg form. False if column is already in the correct form.

checkConvergence

protected int checkConvergence(int workingSize)

Checks for convergence of lower 2x2 sub-matrix within working matrix to upper triangular or block upper triangular form. If convergence is found, this will also zero out the values which have converged to near zero.

Specified by:

checkConvergence in class Schur<Matrix,double[]>

Parameters:

workingSize - Size of current working matrix.

Returns:

Returns the amount the working matrix size should be deflated. Will be zero if no convergence is detected, one if convergence to upper triangular form is detected and two if convergence to block upper triangular form is detected.

real2ComplexSchur

public CMatrix[] real2ComplexSchur()

Converts the real schur form computed in the last decomposition to the complex Schur form.

That is, converts the real block upper triangular Schur matrix to a complex valued properly upper triangular matrix. If the unitary transformation matrix U was computed, the transformations will also be updated accordingly.

This method was adapted from the code given by scipy.linalg.rsf2csf (v1.12.0).

Returns:

An array of length 2 containing the complex Schur matrix T from the last decomposition, and if computed, the complex unitary transformation matrix U from the decomposition. If U was not computed, then the arrays second value will be null.