org.ojalgo.matrix.decomposition

Interface Eigenvalue<N extends Number>

All Superinterfaces:

DeterminantTask<N>, MatrixDecomposition<N>, MatrixDecomposition.Determinant<N>, MatrixDecomposition.Hermitian<N>, MatrixDecomposition.Ordered<N>, MatrixDecomposition.Values<N>, MatrixTask<N>

All Known Implementing Classes:

HermitianEvD

public interface Eigenvalue<N extends Number>
extends MatrixDecomposition<N>, MatrixDecomposition.Hermitian<N>, MatrixDecomposition.Determinant<N>, MatrixDecomposition.Values<N>

[A] = [V][D][V]^-1 ([A][V] = [V][D])

• [A] = any square matrix.
• [V] = contains the eigenvectors as columns.
• [D] = a diagonal matrix with the eigenvalues on the diagonal (possibly in blocks).

[A] is normal if [A][A]^H = [A]^H[A], and [A] is normal if and only if there exists a unitary matrix [Q] such that [A] = [Q][D][Q]^H. Hermitian matrices are normal.

[V] and [D] can always be calculated in the sense that they will satisfy [A][V] = [V][D], but it is not always possible to calculate [V]^-1. (Check the rank and/or the condition number of [V] to determine the validity of [V][D][V]^-1.)

The eigenvalues (and their corresponding eigenvectors) of a non-symmetric matrix could be complex.

Author:

apete

Nested Class Summary

Nested Classes

Modifier and Type	Interface and Description
static class	Eigenvalue.Eigenpair
static interface	Eigenvalue.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 Eigenvalue.Factory<BigDecimal>	BIG
static Eigenvalue.Factory<ComplexNumber>	COMPLEX
static Eigenvalue.Factory<Double>	PRIMITIVE
static Eigenvalue.Factory<Quaternion>	QUATERNION
static Eigenvalue.Factory<RationalNumber>	RATIONAL

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

TYPICAL

Method Summary

Modifier and Type	Method and Description
default void	copyEigenvector(int index, Array1D<ComplexNumber> destination) Deprecated. With Java 9 this will be made private. Use getEigenvectors() or getEigenvector(int) instead.
static <N extends Number> boolean	equals(MatrixStore<N> matrix, Eigenvalue<N> decomposition, NumberContext context)
MatrixStore<N>	getD() The only requirements on [D] are that it should contain the eigenvalues and that [A][V] = [V][D].
default Eigenvalue.Eigenpair	getEigenpair(int index)
Array1D<ComplexNumber>	getEigenvalues() Even for real matrices the eigenvalues (and eigenvectors) are potentially complex numbers.
default void	getEigenvalues(double[] realParts, Optional<double[]> imaginaryParts)
default MatrixStore<ComplexNumber>	getEigenvector(int index) Deprecated. v43 Use getEigenpair(int) instead.
default MatrixStore<ComplexNumber>	getEigenvectors()
ComplexNumber	getTrace() A matrix' trace is the sum of the diagonal elements.
MatrixStore<N>	getV() The columns of [V] represent the eigenvectors of [A] in the sense that [A][V] = [V][D].
boolean	isHermitian() If [A] is hermitian then [V][D][V]-1 becomes [Q][D][Q]H...
boolean	isOrdered() The eigenvalues in D (and the eigenvectors in V) are not necessarily ordered.
static <N extends Number> Eigenvalue<N>	make(Access2D<N> typical)
static <N extends Number> Eigenvalue<N>	make(Access2D<N> typical, boolean hermitian)
default MatrixStore<N>	reconstruct()
static <N extends Number> MatrixStore<N>	reconstruct(Eigenvalue<N> decomposition)

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

checkAndCompute

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

getDeterminant

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

calculateDeterminant

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

computeValuesOnly

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

decompose, isComputed, reset

Field Detail

BIG

static final Eigenvalue.Factory<BigDecimal> BIG

COMPLEX

static final Eigenvalue.Factory<ComplexNumber> COMPLEX

PRIMITIVE

static final Eigenvalue.Factory<Double> PRIMITIVE

QUATERNION

static final Eigenvalue.Factory<Quaternion> QUATERNION

RATIONAL

static final Eigenvalue.Factory<RationalNumber> RATIONAL

Method Detail

make

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

make

static <N extends Number> Eigenvalue<N> make(Access2D<N> typical,
                                             boolean hermitian)

equals

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

reconstruct

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

copyEigenvector

@Deprecated
default void copyEigenvector(int index,
                                         Array1D<ComplexNumber> destination)

Deprecated. With Java 9 this will be made private. Use getEigenvectors() or getEigenvector(int) instead.

getD

MatrixStore<N> getD()

The only requirements on [D] are that it should contain the eigenvalues and that [A][V] = [V][D]. The ordering of the eigenvalues is not specified.

• If [A] is real and symmetric then [D] is (purely) diagonal with real eigenvalues.
• If [A] is real but not symmetric then [D] is block-diagonal with real eigenvalues in 1-by-1 blocks and complex eigenvalues in 2-by-2 blocks.
• If [A] is complex then [D] is (purely) diagonal with complex eigenvalues.

Returns:

The (block) diagonal eigenvalue matrix.

getEigenpair

default Eigenvalue.Eigenpair getEigenpair(int index)

getEigenvalues

Array1D<ComplexNumber> getEigenvalues()

Even for real matrices the eigenvalues (and eigenvectors) are potentially complex numbers. Typically they need to be expressed as complex numbers when [A] is not symmetric.

Prior to v41 this array should always be ordered in descending order - largest (modulus) first. As of v41 the values should be in the same order as the matrices "V" and "D", and if that is ordered or not is indicated by the isOrdered() method.

Returns:

The eigenvalues.

getEigenvalues

default void getEigenvalues(double[] realParts,
                            Optional<double[]> imaginaryParts)

Parameters:

realParts - An array that will receive the real parts of the eigenvalues

imaginaryParts - An optional array that, if present, will receive the imaginary parts of the eigenvalues

getEigenvector

@Deprecated
default MatrixStore<ComplexNumber> getEigenvector(int index)

Deprecated. v43 Use getEigenpair(int) instead.

Parameters:

index - Index corresponding to an entry in getEigenvalues() and/or a column in getEigenvectors().

Returns:

One eigenvector

getEigenvectors

default MatrixStore<ComplexNumber> getEigenvectors()

Returns:

A complex valued alternative to getV().

getTrace

ComplexNumber getTrace()

A matrix' trace is the sum of the diagonal elements. It is also the sum of the eigenvalues. This method should return the sum of the eigenvalues.

Returns:

The matrix' trace

getV

MatrixStore<N> getV()

The columns of [V] represent the eigenvectors of [A] in the sense that [A][V] = [V][D].

Returns:

The eigenvector matrix.

isHermitian

boolean isHermitian()

If [A] is hermitian then [V][D][V]^-1 becomes [Q][D][Q]^H...

isOrdered

boolean isOrdered()

The eigenvalues in D (and the eigenvectors in V) are not necessarily ordered. This is a property of the algorithm/implementation, not the data.

Specified by:

isOrdered in interface MatrixDecomposition.Ordered<N extends Number>

Returns:

true if they are ordered

reconstruct

default MatrixStore<N> reconstruct()

Specified by:

reconstruct in interface MatrixDecomposition<N extends Number>