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>

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.RankRevealing<N extends Number>, MatrixDecomposition.Solver<N extends Number>, MatrixDecomposition.Values<N extends Number>

Field Summary

Fields

Modifier and Type	Field and Description
static SingularValue.Factory<BigDecimal>	BIG
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)

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, setFullSize

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

getRank, 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

Field Detail

BIG

static final SingularValue.Factory<BigDecimal> BIG

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

make

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

equals

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

reconstruct

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

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>