Interface Eigenvalue<N extends Comparable<N>>
- All Superinterfaces:
DeterminantTask<N>
,MatrixDecomposition<N>
,MatrixDecomposition.Determinant<N>
,MatrixDecomposition.Hermitian<N>
,MatrixDecomposition.Ordered<N>
,MatrixDecomposition.Values<N>
,MatrixTask<N>
,Provider2D
,Provider2D.Determinant<N>
,Provider2D.Eigenpairs
,Structure1D
,Structure2D
- All Known Subinterfaces:
Eigenvalue.Generalised<N>
- All Known Implementing Classes:
DynamicEvD
,DynamicEvD.R064
,EigenvalueDecomposition
,GeneralEvD
,GeneralEvD.R064
,GeneralisedEvD
,HermitianEvD
,HermitianEvD.C128
,HermitianEvD.H256
,HermitianEvD.Q128
,HermitianEvD.R064
,HermitianEvD.R128
,RawEigenvalue
,RawEigenvalue.Dynamic
,RawEigenvalue.General
,RawEigenvalue.Symmetric
- [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.
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Nested Class Summary
Nested ClassesModifier and TypeInterfaceDescriptionstatic class
static interface
Eigenvalue.Factory<N extends Comparable<N>>
static enum
static interface
Eigenvalue.Generalised<N extends Comparable<N>>
Nested classes/interfaces inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition
MatrixDecomposition.Determinant<N extends Comparable<N>>, MatrixDecomposition.EconomySize<N extends Comparable<N>>, MatrixDecomposition.Hermitian<N extends Comparable<N>>, MatrixDecomposition.Ordered<N extends Comparable<N>>, MatrixDecomposition.Pivoting<N extends Comparable<N>>, MatrixDecomposition.RankRevealing<N extends Comparable<N>>, MatrixDecomposition.Solver<N extends Comparable<N>>, MatrixDecomposition.Values<N extends Comparable<N>>
Nested classes/interfaces inherited from interface org.ojalgo.matrix.Provider2D
Provider2D.Condition, Provider2D.Determinant<N extends Comparable<N>>, Provider2D.Eigenpairs, Provider2D.Hermitian, Provider2D.Inverse<M>, Provider2D.Rank, Provider2D.Solution<M>, Provider2D.Symmetric, Provider2D.Trace<N extends Comparable<N>>
Nested classes/interfaces inherited from interface org.ojalgo.structure.Structure1D
Structure1D.BasicMapper<T>, Structure1D.IndexMapper<T>, Structure1D.IntIndex, Structure1D.LongIndex, Structure1D.LoopCallback
Nested classes/interfaces inherited from interface org.ojalgo.structure.Structure2D
Structure2D.IntRowColumn, Structure2D.Logical<S extends Structure2D,
B extends Structure2D.Logical<S, B>>, Structure2D.LongRowColumn, Structure2D.ReducibleTo1D<R extends Structure1D>, Structure2D.Reshapable, Structure2D.RowColumnKey<R, C>, Structure2D.RowColumnMapper<R, C> -
Field Summary
FieldsModifier and TypeFieldDescriptionstatic final Eigenvalue.Factory
<ComplexNumber> static final Comparator
<ComplexNumber> Sorts on the norm in descending order.static final Eigenvalue.Factory
<Quaternion> static final Eigenvalue.Factory
<RationalNumber> static final Eigenvalue.Factory
<Double> static final Eigenvalue.Factory
<Quadruple> Fields inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition
TYPICAL
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Method Summary
Modifier and TypeMethodDescriptionprivate void
copyEigenvector
(int index, Array1D<ComplexNumber> destination) static <N extends Comparable<N>>
booleanequals
(MatrixStore<N> matrix, Eigenvalue<N> decomposition, NumberContext context) 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) default List
<Eigenvalue.Eigenpair> This list is always ordered in descending eigenvalue order – that's regardless of ifisOrdered()
returns true or false.Even for real matrices the eigenvalues (and eigenvectors) are potentially complex numbers.default void
getEigenvalues
(double[] realParts, Optional<double[]> imaginaryParts) default MatrixStore
<ComplexNumber> getTrace()
A matrix' trace is the sum of the diagonal elements.getV()
The columns of [V] represent the eigenvectors of [A] in the sense that [A][V] = [V][D].boolean
If [A] is hermitian then [V][D][V]-1 becomes [Q][D][Q]H...boolean
The eigenvalues in D (and the eigenvectors in V) are not necessarily ordered.default MatrixStore
<N> Methods inherited from interface org.ojalgo.matrix.task.DeterminantTask
calculateDeterminant
Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition
decompose, isComputed, reset
Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.Determinant
getDeterminant, toDeterminantProvider
Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.Hermitian
checkAndDecompose
Methods inherited from interface org.ojalgo.matrix.decomposition.MatrixDecomposition.Values
computeValuesOnly
Methods inherited from interface org.ojalgo.structure.Structure2D
count, countColumns, countRows, firstInColumn, firstInRow, getColDim, getMaxDim, getMinDim, getRowDim, isEmpty, isFat, isScalar, isSquare, isTall, isVector, limitOfColumn, limitOfRow, size
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Field Details
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C128
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DESCENDING_NORM
Sorts on the norm in descending order. If the 2 eigenvalues have equal norm then the usualComplexNumber
sort order is used (reversed). -
H256
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Q128
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R064
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R128
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Method Details
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equals
static <N extends Comparable<N>> boolean equals(MatrixStore<N> matrix, Eigenvalue<N> decomposition, NumberContext context) -
copyEigenvector
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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.
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getEigenpair
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getEigenpairs
This list is always ordered in descending eigenvalue order – that's regardless of ifisOrdered()
returns true or false.- Specified by:
getEigenpairs
in interfaceProvider2D.Eigenpairs
- See Also:
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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.
The values should be in the same order as the matrices "V" and "D", and if they is ordered or not is indicated by the
isOrdered()
method.- Returns:
- The eigenvalues.
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getEigenvalues
- Parameters:
realParts
- An array that will receive the real parts of the eigenvaluesimaginaryParts
- An optional array that, if present, will receive the imaginary parts of the eigenvalues
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getEigenvectors
- Returns:
- A complex valued alternative to
getV()
.
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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
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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.
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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 interfaceMatrixDecomposition.Ordered<N extends Comparable<N>>
- Returns:
- true if they are ordered
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reconstruct
- Specified by:
reconstruct
in interfaceMatrixDecomposition<N extends Comparable<N>>
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