com.numericalmethod.suanshu.matrix.doubles.factorization.eigen.qr

Class Hessenberg

java.lang.Object

com.numericalmethod.suanshu.matrix.doubles.factorization.eigen.qr.Hessenberg

All Implemented Interfaces:

java.io.Serializable

public class Hessenberg
extends java.lang.Object
implements java.io.Serializable

An upper Hessenberg matrix is a square matrix which has zero entries below the first sub-diagonal. For example, \[ \begin{bmatrix} 1 & 2 & 3 & 4 & \\ 5 & 6 & 7 & 8 & \\ 0 & 9 & 10 & 11 & \\ 0 & 0 & 12 & 13 & \end{bmatrix} \]

See Also:

Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	Hessenberg.DefaultDeflationCriterion The default deflation criterion is to use eq.
static class	Hessenberg.Deflation This class encapsulates the indices for the upper left hand corner and lower right hand corner of H22 as a result of the deflation procedure.
static interface	Hessenberg.DeflationCriterion Deflation of an upper Hessenberg matrix splits it into multiple smaller upper Hessenberg matrices when the sub-diagonal entries are sufficiently small.

Field Summary

Fields

Modifier and Type	Field and Description
Hessenberg.DeflationCriterion	deflationCriterion

Constructor Summary

Constructors

Constructor and Description
Hessenberg() Construct a Hessenberg utility class with the default deflation criterion.
Hessenberg(Hessenberg.DeflationCriterion dc) Construct a Hessenberg utility class.

Method Summary

Modifier and Type	Method and Description
Hessenberg.Deflation	backSearch(Matrix H, int n, double epsilon) Find H22 such that H22 is the largest unreduced Hessenberg sub-matrix.
static boolean	isHessenberg(Matrix H, double epsilon) Check if H is upper Hessenberg.
boolean	isReducible(Matrix H, double epsilon) Check if H is upper Hessenberg and is reducible.
java.util.ArrayList<Matrix>	reduce(Matrix H) Deprecated. Not supported yet.

Methods inherited from class java.lang.Object

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

Field Detail

deflationCriterion

public final Hessenberg.DeflationCriterion deflationCriterion

Constructor Detail

Hessenberg

public Hessenberg(Hessenberg.DeflationCriterion dc)

Construct a Hessenberg utility class.

Parameters:

dc - a deflation criterion

Hessenberg

public Hessenberg()

Construct a Hessenberg utility class with the default deflation criterion.

Method Detail

isHessenberg

public static boolean isHessenberg(Matrix H,
                                   double epsilon)

Check if H is upper Hessenberg. An upper Hessenberg matrix has zero entries below the first sub-diagonal.

Parameters:

H - a matrix

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

Returns:

true if H is upper Hessenberg

isReducible

public boolean isReducible(Matrix H,
                           double epsilon)

Check if H is upper Hessenberg and is reducible. Note that matrix |0| is unreducible.

Parameters:

H - a matrix

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

Returns:

true if H is upper Hessenberg and is reducible

reduce

@Deprecated
public java.util.ArrayList<Matrix> reduce(Matrix H)

Deprecated. Not supported yet.

This looks for 0s in the sub-diagonal of a Hessenberg matrix, H, and reduces (splits) it into a number of non-reducible Hessenberg matrices. For example, we reduce \[ \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6\\ 7 & 8 & 9 & 10 & 11 & 12\\ 0 & 14 & 15 & 16 & 17 & 18 \\ 0 & 0 & 0 & 22 & 23 & 24 \\ 0 & 0 & 0 & 28 & 29 & 30 \\ 0 & 0 & 0 & 0 & 35 & 36 \end{matrix} \] into \[ \begin{matrix} 1 & 2 & 3 \\ 7 & 8 & 9 \\ 0 & 14 & 15 \\ \end{matrix} \] and \[ \begin{matrix} 22 & 23 & 24 \\ 28 & 29 & 30 \\ 0 & 35 & 36 \\ \end{matrix} \] Not yet implemented.

Parameters:

H - a matrix

Returns:

a list of non-reducible Hessenberg matrices

backSearch

public Hessenberg.Deflation backSearch(Matrix H,
                                       int n,
                                       double epsilon)

Find H₂₂ such that H₂₂ is the largest unreduced Hessenberg sub-matrix. For performance reason, this implementation has a side effect: the method modifies the input H by rounding the negligible sub-diagonal elements to 0.

Parameters:

H - a Hessenberg matrix

n - the start index for the back search

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

Returns:

deflation information that specifies the upper left hand corner and lower right hand corner indices of H₂₂

See Also:

"G. W. Steward, "Algorithm 3.3," Matrix Algorithms, Volume II"