SvdImplicitQrAlgorithm (Efficient Java Matrix Library 0.23 API)

java.lang.Object
- org.ejml.alg.dense.decomposition.svd.implicitqr.SvdImplicitQrAlgorithm

```
public class SvdImplicitQrAlgorithm
extends Object
```
Computes the QR decomposition of a bidiagonal matrix. Internally this matrix is stored as two arrays. Shifts can either be provided to it or it can generate the shifts on its own. It optionally computes the U and V matrices. This comparability allows it to be used to compute singular values and associated matrices efficiently.

A = U*S*V^T
where A is the original m by n matrix.

Based off of the outline provided in:

David S. Watkins, "Fundamentals of Matrix Computations," Second Edition. Page 404-411

Note: To watch it process the matrix step by step uncomment commented out code.

Author:

Peter Abeles

Field Summary

Fields
Modifier and Type	Field and Description
`protected double[]`	`diag`
`protected EigenvalueSmall`	`eigenSmall`
`protected double`	`maxValue`
`protected int`	`N`
`protected int`	`nextExceptional`
`protected int`	`numExceptional`
`protected int`	`numSplits`
`protected double[]`	`off`
`protected Random`	`rand`
`protected int[]`	`splits`
`protected int`	`totalSteps`
`protected DenseMatrix64F`	`Ut`
`protected DenseMatrix64F`	`Vt`
`protected int`	`x1`
`protected int`	`x2`

Constructor Summary

Constructors
Constructor and Description

SvdImplicitQrAlgorithm()

SvdImplicitQrAlgorithm(boolean fastValues)

Constructors
Constructor and Description
`SvdImplicitQrAlgorithm()`
`SvdImplicitQrAlgorithm(boolean fastValues)`

Method Summary

Methods
Modifier and Type	Method and Description
`boolean`	`_process()`
`protected boolean`	`checkForAndHandleZeros()` Checks to see if either the diagonal element or off diagonal element is zero.
`protected void`	`computeRotator(double rise, double run)` Computes a rotator that will set run to zero (?)
`protected void`	`createBulge(int x1, double p, double scale, boolean byAngle)` Performs a similar transform on B^TB-pI
`protected void`	`eigenBB_2x2(int x1)` Computes the eigenvalue of the 2 by 2 matrix B^TB
`void`	`exceptionShift()` It is possible for the QR algorithm to get stuck in a loop because of symmetries.
`double[]`	`getDiag()`
`double`	`getMaxValue()`
`int`	`getNumberOfSingularValues()`
`double[]`	`getOff()`
`double`	`getSingularValue(int index)`
`double[]`	`getSingularValues()`
`DenseMatrix64F`	`getUt()`
`DenseMatrix64F`	`getVt()`
`void`	`incrementSteps()`
`void`	`initParam(int M, int N)`
`boolean`	`isDiagonalZero(int i)`
`boolean`	`isOffZero(int i)`
`boolean`	`nextSplit()` Tells it to process the submatrix at the next split.
`void`	`performImplicitSingleStep(double scale, double lambda, boolean byAngle)` Given the lambda value perform an implicit QR step on the matrix.
`void`	`printMatrix()`
`boolean`	`process()`
`boolean`	`process(double[] values)` Perform a sequence of steps based off of the singular values provided.
`protected void`	`removeBulgeLeft(int x1, boolean notLast)`
`protected void`	`removeBulgeRight(int x1)`
`void`	`resetSteps()`
`double`	`selectWilkinsonShift(double scale)` Selects the Wilkinson's shift for B^TB.
`void`	`setFastValues(boolean b)`
`void`	`setMatrix(int numRows, int numCols, double[] diag, double[] off)`
`void`	`setMaxValue(double maxValue)`
`void`	`setSubmatrix(int x1, int x2)`
`void`	`setUt(DenseMatrix64F ut)`
`void`	`setVt(DenseMatrix64F vt)`
`double[]`	`swapDiag(double[] diag)`
`double[]`	`swapOff(double[] off)`
`protected void`	`updateRotator(DenseMatrix64F Q, int m, int n, double c, double s)` Multiplied a transpose orthogonal matrix Q by the specified rotator.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

rand
```
protected Random rand
```

Ut
```
protected DenseMatrix64F Ut
```

Vt
```
protected DenseMatrix64F Vt
```

totalSteps
```
protected int totalSteps
```

maxValue
```
protected double maxValue
```

N
```
protected int N
```

eigenSmall
```
protected EigenvalueSmall eigenSmall
```

numExceptional
```
protected int numExceptional
```

nextExceptional
```
protected int nextExceptional
```

diag
```
protected double[] diag
```

off
```
protected double[] off
```

x1
```
protected int x1
```

x2
```
protected int x2
```

splits
```
protected int[] splits
```

numSplits
```
protected int numSplits
```

Constructor Detail

SvdImplicitQrAlgorithm

public SvdImplicitQrAlgorithm(boolean fastValues)

SvdImplicitQrAlgorithm
```
public SvdImplicitQrAlgorithm()
```

Method Detail

getUt
```
public DenseMatrix64F getUt()
```

setUt
```
public void setUt(DenseMatrix64F ut)
```

getVt
```
public DenseMatrix64F getVt()
```

setVt
```
public void setVt(DenseMatrix64F vt)
```

setMatrix

public void setMatrix(int numRows,
             int numCols,
             double[] diag,
             double[] off)

swapDiag

public double[] swapDiag(double[] diag)

swapOff

public double[] swapOff(double[] off)

setMaxValue

public void setMaxValue(double maxValue)

initParam

public void initParam(int M,
             int N)

process
```
public boolean process()
```

process
```
public boolean process(double[] values)
```
Perform a sequence of steps based off of the singular values provided.

Parameters:
values -

Returns:

_process
```
public boolean _process()
```

incrementSteps
```
public void incrementSteps()
```

isOffZero
```
public boolean isOffZero(int i)
```

isDiagonalZero
```
public boolean isDiagonalZero(int i)
```

resetSteps
```
public void resetSteps()
```

nextSplit
```
public boolean nextSplit()
```
Tells it to process the submatrix at the next split. Should be called after the current submatrix has been processed.

performImplicitSingleStep

public void performImplicitSingleStep(double scale,
                             double lambda,
                             boolean byAngle)

Given the lambda value perform an implicit QR step on the matrix. B^T*B-lambda*I

Parameters:: lambda - Stepping factor.

updateRotator
```
protected void updateRotator(DenseMatrix64F Q,
                 int m,
                 int n,
                 double c,
                 double s)
```
Multiplied a transpose orthogonal matrix Q by the specified rotator. This is used to update the U and V matrices. Updating the transpose of the matrix is faster since it only modifies the rows.

Parameters:
Q - Orthogonal matrix
m - Coordinate of rotator.
n - Coordinate of rotator.
c - cosine of rotator.
s - sine of rotator.

createBulge

protected void createBulge(int x1,
               double p,
               double scale,
               boolean byAngle)

Performs a similar transform on B^TB-pI

computeRotator

protected void computeRotator(double rise,
                  double run)

Computes a rotator that will set run to zero (?)

removeBulgeLeft

protected void removeBulgeLeft(int x1,
                   boolean notLast)

removeBulgeRight

protected void removeBulgeRight(int x1)

setSubmatrix

public void setSubmatrix(int x1,
                int x2)

selectWilkinsonShift
```
public double selectWilkinsonShift(double scale)
```
Selects the Wilkinson's shift for B^TB. See page 410. It is guaranteed to converge and converges fast in practice.

Parameters:
scale - Scale factor used to help prevent overflow/underflow

Returns:
Shifting factor lambda/(scale*scale)

eigenBB_2x2
```
protected void eigenBB_2x2(int x1)
```
Computes the eigenvalue of the 2 by 2 matrix B^TB

checkForAndHandleZeros
```
protected boolean checkForAndHandleZeros()
```
Checks to see if either the diagonal element or off diagonal element is zero. If one is then it performs a split or pushes it off the matrix.

Returns:
True if there was a zero.

exceptionShift
```
public void exceptionShift()
```
It is possible for the QR algorithm to get stuck in a loop because of symmetries. This happens more often with larger matrices. By taking a random step it can break the symmetry and finish.

printMatrix
```
public void printMatrix()
```

getNumberOfSingularValues

public int getNumberOfSingularValues()

getSingularValue

public double getSingularValue(int index)

setFastValues
```
public void setFastValues(boolean b)
```

getSingularValues
```
public double[] getSingularValues()
```

getDiag
```
public double[] getDiag()
```

getOff
```
public double[] getOff()
```

getMaxValue
```
public double getMaxValue()
```

Class SvdImplicitQrAlgorithm

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

rand

Ut

Vt

totalSteps

maxValue

N

eigenSmall

numExceptional

nextExceptional

diag

off

x1

x2

splits

numSplits

Constructor Detail

SvdImplicitQrAlgorithm

SvdImplicitQrAlgorithm

Method Detail

getUt

setUt

getVt

setVt

setMatrix

swapDiag

swapOff

setMaxValue

initParam

process

process

_process

incrementSteps

isOffZero

isDiagonalZero

resetSteps

nextSplit

performImplicitSingleStep

updateRotator

createBulge

computeRotator

removeBulgeLeft

removeBulgeRight

setSubmatrix

selectWilkinsonShift

eigenBB_2x2

checkForAndHandleZeros

exceptionShift

printMatrix

getNumberOfSingularValues

getSingularValue

setFastValues

getSingularValues

getDiag

getOff

getMaxValue