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JMSLTM Numerical Library 4.0 | ||||||||
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java.lang.Object com.imsl.math.Eigen
Collection of Eigen System functions.
Eigen
computes the eigenvalues and eigenvectors of a real
matrix. The matrix is first balanced. Orthogonal similarity transformations
are used to reduce the balanced matrix to a real upper Hessenberg matrix.
The implicit double-shifted QR algorithm is used to compute the eigenvalues
and eigenvectors of this Hessenberg matrix. The eigenvectors are normalized
such that each has Euclidean length of value one. The largest component is
real and positive.
The balancing routine is based on the EISPACK routine BALANC
.
The reduction routine is based on the EISPACK routines ORTHES
and
ORTRAN
. The QR algorithm routine is based on the EISPACK routine
HQR2
. See Smith et al. (1976) for the EISPACK routines. Further
details, some timing data, and credits are given in Hanson et al. (1990).
While the exact value of the performance index, ,
is highly machine dependent, the performance of Eigen
is
considered excellent if , good if
, and poor if
.
The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124-125).
Nested Class Summary | |
static class |
Eigen.DidNotConvergeException
The iteration did not converge |
Constructor Summary | |
Eigen(double[][] a)
Constructs the eigenvalues and the eigenvectors of a real square matrix. |
|
Eigen(double[][] a,
boolean computeVectors)
Constructs the eigenvalues and (optionally) the eigenvectors of a real square matrix. |
Method Summary | |
Complex[] |
getValues()
Returns the eigenvalues of a matrix of type double . |
Complex[][] |
getVectors()
Returns the eigenvectors. |
double |
performanceIndex(double[][] a)
Returns the performance index of a real eigensystem. |
Methods inherited from class java.lang.Object |
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Constructor Detail |
public Eigen(double[][] a) throws Eigen.DidNotConvergeException
a
- is the double
square matrix whose eigensystem is to be
constructed
Eigen.DidNotConvergeException
- is thrown when the algorithm
fails to converge on the eigenvalues of the matrix.public Eigen(double[][] a, boolean computeVectors) throws Eigen.DidNotConvergeException
a
- is the double
square matrix whose eigensystem is to be
constructedcomputeVectors
- is true if the eigenvectors are to be computed
Eigen.DidNotConvergeException
- is thrown when the algorithm
fails to converge on the eigenvalues of the matrix.Method Detail |
public Complex[] getValues()
double
.
Complex
array containing the eigenvalues
of this matrix in descending orderpublic Complex[][] getVectors()
Complex
matrix containing the eigenvectors. The eigenvector
corresponding to the j-th eigenvalue is stored in the
j-th column.
Each vector is normalized to have Euclidean length one.public double performanceIndex(double[][] a)
a
- a double
matrix
double
scalar value indicating how well the algorithms
which have computed the eigenvalue and eigenvector pairs
have performed. A performance index less than 1 is
considered excellent, 1 to 100 is good, while greater
than 100 is considered poor.
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