org.ojalgo.matrix.decomposition

Interface SingularValue<N extends Number>

All Superinterfaces:

InverterTask<N>, MatrixDecomposition<N>, MatrixDecomposition.EconomySize<N>, MatrixDecomposition.Ordered<N>, MatrixDecomposition.RankRevealing<N>, MatrixDecomposition.Solver<N>, MatrixDecomposition.Values<N>, MatrixTask<N>, SolverTask<N>, Structure1D, Structure2D

public interface SingularValue<N extends Number>
extends MatrixDecomposition<N>, MatrixDecomposition.Solver<N>, MatrixDecomposition.EconomySize<N>, MatrixDecomposition.RankRevealing<N>, MatrixDecomposition.Values<N>

Singular Value: [A] = [Q1][D][Q2]^T Decomposes [this] into [Q1], [D] and [Q2] where:

• [Q1] is an orthogonal matrix. The columns are the left, orthonormal, singular vectors of [this]. Its columns are the eigenvectors of [A][A]^T, and therefore has the same number of rows as [this].
• [D] is a diagonal matrix. The elements on the diagonal are the singular values of [this]. It is either square or has the same dimensions as [this]. The singular values of [this] are the square roots of the nonzero eigenvalues of [A][A]^T and [A]^T[A] (they are the same)
• [Q2] is an orthogonal matrix. The columns are the right, orthonormal, singular vectors of [this]. Its columns are the eigenvectors of [A][A]^T, and therefore has the same number of rows as [this] has columns.
• [this] = [Q1][D][Q2]^T

A singular values decomposition always exists.

Author:

apete

Nested Class Summary

Nested Classes

Modifier and Type	Interface and Description
static interface	SingularValue.Factory<N extends Number>

Nested classes/interfaces inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition

MatrixDecomposition.Determinant<N extends Number>, MatrixDecomposition.EconomySize<N extends Number>, MatrixDecomposition.Hermitian<N extends Number>, MatrixDecomposition.Ordered<N extends Number>, MatrixDecomposition.Pivoting<N extends Number>, MatrixDecomposition.RankRevealing<N extends Number>, MatrixDecomposition.Solver<N extends Number>, MatrixDecomposition.Values<N extends Number>

Nested classes/interfaces inherited from interface org.ojalgo.structure.Structure2D

Structure2D.IntRowColumn, Structure2D.Logical<S extends Structure2D,B extends Structure2D.Logical<S,?>>, Structure2D.LongRowColumn, Structure2D.ReducibleTo1D<R extends Structure1D>, Structure2D.RowColumnCallback, Structure2D.RowColumnKey<R,C>, Structure2D.RowColumnMapper<R,C>

Nested classes/interfaces inherited from interface org.ojalgo.structure.Structure1D

Structure1D.BasicMapper<T>, Structure1D.IndexCallback, Structure1D.IndexMapper<T>, Structure1D.IntIndex, Structure1D.LongIndex, Structure1D.LoopCallback

Field Summary

Fields

Modifier and Type	Field and Description
static SingularValue.Factory<ComplexNumber>	COMPLEX
static SingularValue.Factory<Double>	PRIMITIVE
static SingularValue.Factory<Quaternion>	QUATERNION
static SingularValue.Factory<RationalNumber>	RATIONAL

Fields inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition

TYPICAL

Method Summary

Modifier and Type	Method and Description
static <N extends Number> boolean	equals(MatrixStore<N> matrix, SingularValue<N> decomposition, NumberContext context)
double	getCondition() The condition number.
MatrixStore<N>	getD()
double	getFrobeniusNorm() Sometimes also called the Schatten 2-norm or Hilbert-Schmidt norm.
double	getKyFanNorm(int k) Ky Fan k-norm.
double	getOperatorNorm()
MatrixStore<N>	getQ1() If [A] is m-by-n and its rank is r, then: The first r columns of [Q1] span the column space, range or image of [A]. The last m-r columns of [Q1] span the left nullspace or cokernel of [A]. Calculating the QR decomposition of [A] is a faster alternative.
MatrixStore<N>	getQ2() If [A] is m-by-n and its rank is r, then: The first r columns of [Q2] span the row space or coimage of [A]. The last n-r columns of [Q2] span the nullspace or kernel of [A]. Calculating the QR decomposition of [A]T is a faster alternative.
Array1D<Double>	getSingularValues()
default void	getSingularValues(double[] values)
double	getTraceNorm()
static <N extends Number> SingularValue<N>	make(Access2D<N> typical)
default MatrixStore<N>	reconstruct()
static <N extends Number> MatrixStore<N>	reconstruct(SingularValue<N> decomposition) Deprecated. v48 Use reconstruct() instead

Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.Solver

compute, getInverse, getInverse, getSolution, getSolution, isSolvable

Methods inherited from interface org.ojalgo.matrix.task.SolverTask

preallocate, preallocate, solve, solve

Methods inherited from interface org.ojalgo.matrix.task.InverterTask

invert, invert, preallocate, preallocate

Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.EconomySize

isFullSize

Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.RankRevealing

countSignificant, getRank, getRankThreshold, isFullRank

Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.Values

computeValuesOnly

Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.Ordered

isOrdered

Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition

decompose, isComputed, reset

Methods inherited from interface org.ojalgo.structure.Structure2D

column, column, column, column, column, count, count, countColumns, countRows, index, index, isEmpty, isFat, isScalar, isSquare, isTall, isVector, loopAll, loopColumn, loopColumn, loopDiagonal, loopMatching, loopRow, loopRow, mapperOf, row, row, row, row, row

Methods inherited from interface org.ojalgo.structure.Structure1D

index, loopAll, loopMatching, loopRange, mapper, size

Field Detail

COMPLEX

static final SingularValue.Factory<ComplexNumber> COMPLEX

PRIMITIVE

static final SingularValue.Factory<Double> PRIMITIVE

QUATERNION

static final SingularValue.Factory<Quaternion> QUATERNION

RATIONAL

static final SingularValue.Factory<RationalNumber> RATIONAL

Method Detail

equals

static <N extends Number> boolean equals(MatrixStore<N> matrix,
                                         SingularValue<N> decomposition,
                                         NumberContext context)

make

static <N extends Number> SingularValue<N> make(Access2D<N> typical)

reconstruct

@Deprecated
static <N extends Number> MatrixStore<N> reconstruct(SingularValue<N> decomposition)

Deprecated. v48 Use reconstruct() instead

getCondition

double getCondition()

The condition number.

Returns:

The largest singular value divided by the smallest singular value.

getD

MatrixStore<N> getD()

Returns:

The diagonal matrix of singular values.

getFrobeniusNorm

double getFrobeniusNorm()

Sometimes also called the Schatten 2-norm or Hilbert-Schmidt norm.

Returns:

The square root of the sum of squares of the singular values.

getKyFanNorm

double getKyFanNorm(int k)

Ky Fan k-norm.

The first Ky Fan k-norm is the operator norm (the largest singular value), and the last is called the trace norm (the sum of all singular values).

Parameters:

k - The number of singular values to add up.

Returns:

The sum of the k largest singular values.

getOperatorNorm

double getOperatorNorm()

Returns:

2-norm

getQ1

MatrixStore<N> getQ1()

If [A] is m-by-n and its rank is r, then:

• The first r columns of [Q1] span the column space, range or image of [A].
• The last m-r columns of [Q1] span the left nullspace or cokernel of [A].

Calculating the QR decomposition of [A] is a faster alternative.

getQ2

MatrixStore<N> getQ2()

If [A] is m-by-n and its rank is r, then:

• The first r columns of [Q2] span the row space or coimage of [A].
• The last n-r columns of [Q2] span the nullspace or kernel of [A].

Calculating the QR decomposition of [A]^T is a faster alternative.

getSingularValues

Array1D<Double> getSingularValues()

Returns:

The singular values ordered in descending order.

getSingularValues

default void getSingularValues(double[] values)

Parameters:

values - An array that will receive the singular values

getTraceNorm

double getTraceNorm()

reconstruct

default MatrixStore<N> reconstruct()

Specified by:

reconstruct in interface MatrixDecomposition<N extends Number>