Class transforming a general real matrix to Schur form.

A m × m matrix A can be written as the product of three matrices: A = P × T × P^T with P an orthogonal matrix and T an quasi-triangular matrix. Both P and T are m × m matrices.

Transformation to Schur form is often not a goal by itself, but it is an intermediate step in more general decomposition algorithms like eigen decomposition. This class is therefore intended for internal use by the library and is not public. As a consequence of this explicitly limited scope, many methods directly returns references to internal arrays, not copies.

This class is based on the method hqr2 in class EigenvalueDecomposition from the JAMA library.

Nested and Inner Type Summary

Modifier and Type	Class and Description
private static class	SchurTransformer.ShiftInfo Internal data structure holding the current shift information.

Field Summary

Modifier and Type	Field and Description
private RealMatrix	cachedP Cached value of P.
private RealMatrix	cachedPt Cached value of PT.
private RealMatrix	cachedT Cached value of T.
private final double	epsilon Epsilon criteria taken from JAMA code (originally was 2^-52).
private final double[][]	matrixP P matrix.
private final double[][]	matrixT T matrix.
private static final int	MAX_ITERATIONS Maximum allowed iterations for convergence of the transformation.

Access	Constructor and Description
pack-priv	SchurTransformer(final RealMatrix matrix to transform matrix) Build the transformation to Schur form of a general real matrix.

Method Summary

Modifier and Type	Method and Description
private void	computeShift(final int the index of the small sub-diagonal element l, final int the current eigenvalue index idx, final int the current iteration iteration, final SchurTransformer.ShiftInfo holder for shift information shift) Compute the shift for the current iteration.
private int	Returns: the index of the first small sub-diagonal element findSmallSubDiagonalElement(final int the starting index for the search startIdx, final double the L1 norm of the matrix norm) Find the first small sub-diagonal element and returns its index.
private double	Returns: the L1 norm of matrix T getNorm() Computes the L1 norm of the (quasi-)triangular matrix T.
public RealMatrix	Returns: the P matrix getP() Returns the matrix P of the transform.
public RealMatrix	Returns: the transpose of the P matrix getPT() Returns the transpose of the matrix P of the transform.
public RealMatrix	Returns: the T matrix getT() Returns the quasi-triangular Schur matrix T of the transform.
private int	Returns: the start index for the QR step initQRStep(int the index of the small sub-diagonal element il, final int the current eigenvalue index iu, final SchurTransformer.ShiftInfo shift information holder shift, double[] the initial houseHolder vector hVec) Initialize the householder vectors for the QR step.
private void	performDoubleQRStep(final int the index of the small sub-diagonal element il, final int the start index for the QR step im, final int the current eigenvalue index iu, final SchurTransformer.ShiftInfo shift information holder shift, final double[] the initial houseHolder vector hVec) Perform a double QR step involving rows l:idx and columns m:n
private void	transform() Transform original matrix to Schur form.

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

Field Detail

cachedP	back to summary
private RealMatrix cachedP Cached value of P.

cachedPt	back to summary
private RealMatrix cachedPt Cached value of PT.

cachedT	back to summary
private RealMatrix cachedT Cached value of T.

epsilon	back to summary
private final double epsilon Epsilon criteria taken from JAMA code (originally was 2^-52).

matrixP	back to summary
private final double[][] matrixP P matrix.

matrixT	back to summary
private final double[][] matrixT T matrix.

MAX_ITERATIONS	back to summary
private static final int MAX_ITERATIONS Maximum allowed iterations for convergence of the transformation.

Constructor Detail

SchurTransformer	back to summary
pack-priv SchurTransformer(final RealMatrix matrix) Build the transformation to Schur form of a general real matrix. Parameters matrix:RealMatrix matrix to transform Exceptions NonSquareMatrixException: if the matrix is not square

SchurTransformer

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pack-priv SchurTransformer(final RealMatrix matrix)

Build the transformation to Schur form of a general real matrix.

Parameters

matrix:RealMatrix: matrix to transform

Exceptions

NonSquareMatrixException:: if the matrix is not square

Method Detail

computeShift	back to summary
private void computeShift(final int l, final int idx, final int iteration, final SchurTransformer.ShiftInfo shift) Compute the shift for the current iteration. Parameters l:int the index of the small sub-diagonal element idx:int the current eigenvalue index iteration:int the current iteration shift:SchurTransformer.ShiftInfo holder for shift information

computeShift

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private void computeShift(final int l, final int idx, final int iteration, final SchurTransformer.ShiftInfo shift)

Compute the shift for the current iteration.

Parameters

l:int: the index of the small sub-diagonal element
idx:int: the current eigenvalue index
iteration:int: the current iteration
shift:SchurTransformer.ShiftInfo: holder for shift information

findSmallSubDiagonalElement	back to summary
private int findSmallSubDiagonalElement(final int startIdx, final double norm) Find the first small sub-diagonal element and returns its index. Parameters startIdx:int the starting index for the search norm:double the L1 norm of the matrix Returns:int the index of the first small sub-diagonal element

findSmallSubDiagonalElement

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private int findSmallSubDiagonalElement(final int startIdx, final double norm)

Find the first small sub-diagonal element and returns its index.

Parameters

startIdx:int: the starting index for the search
norm:double: the L1 norm of the matrix

Returns:int

the index of the first small sub-diagonal element

getNorm	back to summary
private double getNorm() Computes the L1 norm of the (quasi-)triangular matrix T. Returns:double the L1 norm of matrix T

getNorm

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private double getNorm()

Computes the L1 norm of the (quasi-)triangular matrix T.

Returns:double: the L1 norm of matrix T

getP	back to summary
public RealMatrix getP() Returns the matrix P of the transform. P is an orthogonal matrix, i.e. its inverse is also its transpose. Returns:RealMatrix the P matrix

getP

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public RealMatrix getP()

Returns the matrix P of the transform.

P is an orthogonal matrix, i.e. its inverse is also its transpose.

Returns:RealMatrix: the P matrix

getPT	back to summary
public RealMatrix getPT() Returns the transpose of the matrix P of the transform. P is an orthogonal matrix, i.e. its inverse is also its transpose. Returns:RealMatrix the transpose of the P matrix

getPT

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public RealMatrix getPT()

Returns the transpose of the matrix P of the transform.

P is an orthogonal matrix, i.e. its inverse is also its transpose.

Returns:RealMatrix: the transpose of the P matrix

getT	back to summary
public RealMatrix getT() Returns the quasi-triangular Schur matrix T of the transform. Returns:RealMatrix the T matrix

getT

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public RealMatrix getT()

Returns the quasi-triangular Schur matrix T of the transform.

Returns:RealMatrix: the T matrix

initQRStep	back to summary
private int initQRStep(int il, final int iu, final SchurTransformer.ShiftInfo shift, double[] hVec) Initialize the householder vectors for the QR step. Parameters il:int the index of the small sub-diagonal element iu:int the current eigenvalue index shift:SchurTransformer.ShiftInfo shift information holder hVec:double[] the initial houseHolder vector Returns:int the start index for the QR step

initQRStep

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private int initQRStep(int il, final int iu, final SchurTransformer.ShiftInfo shift, double[] hVec)

Initialize the householder vectors for the QR step.

Parameters

il:int: the index of the small sub-diagonal element
iu:int: the current eigenvalue index
shift:SchurTransformer.ShiftInfo: shift information holder
hVec:double[]: the initial houseHolder vector

Returns:int

the start index for the QR step

performDoubleQRStep	back to summary
private void performDoubleQRStep(final int il, final int im, final int iu, final SchurTransformer.ShiftInfo shift, final double[] hVec) Perform a double QR step involving rows l:idx and columns m:n Parameters il:int the index of the small sub-diagonal element im:int the start index for the QR step iu:int the current eigenvalue index shift:SchurTransformer.ShiftInfo shift information holder hVec:double[] the initial houseHolder vector

performDoubleQRStep

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private void performDoubleQRStep(final int il, final int im, final int iu, final SchurTransformer.ShiftInfo shift, final double[] hVec)

Perform a double QR step involving rows l:idx and columns m:n

Parameters

il:int: the index of the small sub-diagonal element
im:int: the start index for the QR step
iu:int: the current eigenvalue index
shift:SchurTransformer.ShiftInfo: shift information holder
hVec:double[]: the initial houseHolder vector

transform	back to summary
private void transform() Transform original matrix to Schur form. Exceptions MaxCountExceededException: if the transformation does not converge

transform

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private void transform()

Transform original matrix to Schur form.

Exceptions

MaxCountExceededException:: if the transformation does not converge

org.apache.commons.math3.linear back to summary

private Class SchurTransformer.ShiftInfo

Class Inheritance

extends Object

java.lang.Object
org.apache.commons.math3.linear.SchurTransformer.ShiftInfo

Internal data structure holding the current shift information. Contains variable names as present in the original JAMA code.

Field Summary

Modifier and Type	Field and Description
pack-priv double	exShift Indicates an exceptional shift.
pack-priv double	w w shift info
pack-priv double	x x shift info
pack-priv double	y y shift info

Constructor Summary

Access	Constructor and Description
private	ShiftInfo()

Method Summary

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

Field Detail

exShift	back to summary
pack-priv double exShift Indicates an exceptional shift.

w	back to summary
pack-priv double w w shift info

x	back to summary
pack-priv double x x shift info

y	back to summary
pack-priv double y y shift info

Constructor Detail

ShiftInfo	back to summary
private ShiftInfo()

pack-priv Class SchurTransformer

Nested and Inner Type Summary

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail

Method Detail

private Class SchurTransformer.ShiftInfo

Field Summary

Constructor Summary

Method Summary

Field Detail

Constructor Detail