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

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.factorization.eigen.Hessenberg

public class Hessenberg
extends java.lang.Object

This class implements the operations for an upper Hessenberg matrix. An upper Hessenberg matrix is a square matrix which has zero entries below the first subdiagonal.

For example,

 | 1 2 3 4   |
 | 5 6 7 8   |
 | 0 9 10 11 |
 | 0 0 12 13 |
 

See Also:

Nested Class Summary

static class	Hessenberg.DefaultDeflationCriterion Users can override the default implementation to define their 0 threshold.
static class	Hessenberg.Deflation This class encapsulates the indices for the upper left hand corner, and lower right hand corner of H22.
static interface	Hessenberg.DeflationCriterion User implements this class to decide whether to split an upper Hessenberg into multiple smaller upper Hessenberg when the sub-diagonal entries are sufficiently small.

Field Summary

Hessenberg.DeflationCriterion

deflationCriterion the deflation criterion used for this instance of Hessenberg

Constructor Summary

Hessenberg()
Create a Hessenberg utility class with the default deflation criterion.

Hessenberg(Hessenberg.DeflationCriterion dc)
Create a Hessenberg utility class with a customized deflation criterion.

Method Summary

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

the deflation criterion used for this instance of Hessenberg

Constructor Detail

Hessenberg

public Hessenberg()

Create a Hessenberg utility class with the default deflation criterion.

Hessenberg

public Hessenberg(Hessenberg.DeflationCriterion dc)

Create a Hessenberg utility class with a customized deflation criterion.

Parameters:

dc - a 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 subdiagonal.

Parameters:

H - a matrix

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

Returns:

true iff H is upper Hessenberg

isReducible

public boolean isReducible(Matrix H,
                           double epsilon)

Check if H is upper Hessenberg and is reducible.

Note: zero matrix

| 0 |

is unreducible.

Parameters:

H - a matrix

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

Returns:

true iff 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. This reduces (splits) H into a number of non-reducible Hessenberg matrices.

For example, we reduce

 |  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 |
 

into

 |  1  2  3 |
 |  7  8  9 |
 |  0 14 15 |
 

and

 | 22 23 24 |
 | 28 29 30 |
 |  0 35 36 |
 

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. A modified version of

 Algorithm 3.3 in
 Matrix Algorithms, Volume II by
 Steward G. W.
 

For performance reason:

Side effect: this modifies H by rounding the negligible sub-diagonal elements to 0

Parameters:

H - the 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 which specifies the upper left hand corner and lower right hand corner indices of H22