Package org.flag4j.linalg.decompositions.svd

Class SVD<T extends MatrixMixin<T,?,?,?,?,?,?,?>>

java.lang.Object

org.flag4j.linalg.decompositions.svd.SVD<T>

Type Parameters:

T - The type of the matrix to compute the singular value decomposition of.

All Implemented Interfaces:

Decomposition<T>

Direct Known Subclasses:

ComplexSVD, RealSVD

public abstract class SVD<T extends MatrixMixin<T,?,?,?,?,?,?,?>>
extends Object
implements Decomposition<T>

This abstract class specifies methods for computing the singular value decomposition (SVD) of a matrix. That is, decompose a rectangular matrix M as M=USV^H where U and V are unitary matrices whose columns are the left and right singular vectors of M and S is a rectangular diagonal matrix containing the singular values of M.

Field Summary

Fields

Modifier and Type	Field	Description
protected boolean	computeUV	Flag which indicates if the singular vectors should be computed in addition to the singular values.
protected int	rank	The rank of the matrix being decomposed.
protected boolean	reduced	Flag which indicates if the reduced (or full) SVD should be computed.
protected Matrix	S	The rectangular diagonal S corresponding to M=USVH in the SVD.
protected T	U	The unitary matrix U corresponding to M=USVH in the SVD.
protected T	V	The unitary matrix V corresponding to M=USVH in the SVD.

Constructor Summary

Constructors

Modifier	Constructor	Description
protected	SVD(boolean computeUV, boolean reduced)	Creates a decomposer to compute the Schur decomposition.

Method Summary

Modifier and Type	Method	Description
protected void	computeRank(int rows, int cols, double[] singularValues)	Computes the rank of the matrix being decomposed using the singular values of the matrix.
SVD<T>	decompose(T src)	Applies decomposition to the source matrix.
protected abstract void	extractNormalizedCols(T singularVecs, int j)	Extracts the singular vectors, normalizes them and sets the columns of U and V to be the left/right singular vectors.
int	getRank()	Gets the rank of the last matrix decomposed.
Matrix	getS()	Gets the diagonal matrix S corresponding to M=USVH in the SVD.
T	getU()	Gets the unitary matrix U corresponding to M=USVH in the SVD.
T	getV()	Gets the unitary matrix V corresponding to M=USVH in the SVD.
protected abstract void	initUV(Shape src, int cols)	Initializes the unitary U and V matrices for the SVD.
protected abstract T	invDirectSum(T src)	Computes the inverse direct sum of a matrix and its hermitian transpose.
protected abstract T	makeEigenPairs(T B, double[] eigVals)	Gets the eigen values and vectors of symmetric block matrix which corresponds to the singular values and vectors of the matrix being decomposed.
protected abstract void	makeEigenVals(T B, double[] eigVals)	Gets the eigen values of the symmetric block matrix which corresponds to the singular values of the matrix being decomposed.

Methods inherited from class java.lang.Object

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

Field Details

computeUV

protected boolean computeUV

Flag which indicates if the singular vectors should be computed in addition to the singular values.

reduced

protected boolean reduced

Flag which indicates if the reduced (or full) SVD should be computed.

U

protected T extends MatrixMixin<T,?,?,?,?,?,?,?> U

The unitary matrix U corresponding to M=USV^H in the SVD.

S

protected Matrix S

The rectangular diagonal S corresponding to M=USV^H in the SVD.

V

protected T extends MatrixMixin<T,?,?,?,?,?,?,?> V

The unitary matrix V corresponding to M=USV^H in the SVD.

rank

protected int rank

The rank of the matrix being decomposed. This is calculated as a byproduct of the decomposition.

Constructor Details

SVD

protected SVD(boolean computeUV,
 boolean reduced)

Creates a decomposer to compute the Schur decomposition.

Parameters:

computeUV - A flag which indicates if the unitary matrices Q and V should be computed (i.e. the singular vectors).
- If true, the Q and V matrices will be computed. - If false, the Q and V matrices will not be computed. If it is not needed, this may provide a performance improvement.

reduced - Flag which indicates if the reduced (or full) SVD should be computed.
- If true, reduced SVD is computed. - If false, the full SVD is computed.

Method Details

getU

public T getU()

Gets the unitary matrix U corresponding to M=USV^H in the SVD.

Returns:

U corresponding to M=USV^H in the SVD.

getS

public Matrix getS()

Gets the diagonal matrix S corresponding to M=USV^H in the SVD.

Returns:

S corresponding to M=USV^H in the SVD.

getV

public T getV()

Gets the unitary matrix V corresponding to M=USV^H in the SVD.

Returns:

V corresponding to M=USV^H in the SVD. Note that the hermitian transpose has not been computed.

getRank

public int getRank()

Gets the rank of the last matrix decomposed. This is computed as a byproduct of the decomposition.

Returns:

The rank of the last matrix decomposed.

decompose

public SVD<T> decompose(T src)

Applies decomposition to the source matrix.

Specified by:

decompose in interface Decomposition<T extends MatrixMixin<T,?,?,?,?,?,?,?>>

Parameters:

src - The source matrix to decompose.

Returns:

A reference to this decomposer.

invDirectSum

protected abstract T invDirectSum(T src)

Computes the inverse direct sum of a matrix and its hermitian transpose.

Parameters:

src - Matrix to inverse direct add with its hermitian transpose.

Returns:

The inverse direct sum of the src matrix with its hermitian transpose.

computeRank

protected void computeRank(int rows,
 int cols,
 double[] singularValues)

Computes the rank of the matrix being decomposed using the singular values of the matrix.

Parameters:

rows - The number of rows in the original source matrix.

cols - The number of columns in the original source matrix.

singularValues - The singular values of the original source matrix.

makeEigenPairs

protected abstract T makeEigenPairs(T B,
 double[] eigVals)

Gets the eigen values and vectors of symmetric block matrix which corresponds to the singular values and vectors of the matrix being decomposed.

Parameters:

B - Symmetric block matrix to compute the eigenvalues of.

eigVals - Storage for eigenvalues.

Returns:

The eigenvalues and eigenvectors of the symmetric block matrix which corresponds to the singular values and vectors of the matrix being decomposed.

makeEigenVals

protected abstract void makeEigenVals(T B,
 double[] eigVals)

Gets the eigen values of the symmetric block matrix which corresponds to the singular values of the matrix being decomposed.

Parameters:

B - Symmetric block matrix to compute the eigenvalues of.

eigVals - Storage for eigenvalues.

initUV

protected abstract void initUV(Shape src,
 int cols)

Initializes the unitary U and V matrices for the SVD.

Parameters:

src - Shape of the source matrix being decomposed.

cols - The number of columns for U and V.

extractNormalizedCols

protected abstract void extractNormalizedCols(T singularVecs,
 int j)

Extracts the singular vectors, normalizes them and sets the columns of U and V to be the left/right singular vectors.

Parameters:

singularVecs - Computed left and right singular vectors.

j - Index of the column of U and V to set.