f08jxf computes the eigenvectors of a real symmetric tridiagonal matrix corresponding to specified eigenvalues, by inverse iteration, storing the eigenvectors in a complex array.
The routine may be called by the names f08jxf, nagf_lapackeig_zstein or its LAPACK name zstein.
3Description
f08jxf computes the eigenvectors of a real symmetric tridiagonal matrix corresponding to specified eigenvalues, by inverse iteration (see Jessup and Ipsen (1992)). It is designed to be used in particular after the specified eigenvalues have been computed by f08jjf with , but may also be used when the eigenvalues have been computed by other routines in Chapters F02 or F08.
The eigenvectors of are real, but are stored by this routine in a complex array. If has been formed by reduction of a full complex Hermitian matrix to tridiagonal form, then eigenvectors of may be transformed to (complex) eigenvectors of by a call to f08fuforf08guf.
f08jjf determines whether the matrix splits into block diagonal form:
and passes details of the block structure to this routine in the arrays iblock and isplit. This routine can then take advantage of the block structure by performing inverse iteration on each block separately, which is more efficient than using the whole matrix.
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Jessup E and Ipsen I C F (1992) Improving the accuracy of inverse iteration SIAM J. Sci. Statist. Comput.13 550–572
5Arguments
1: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
2: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
3: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array e
must be at least
.
On entry: the off-diagonal elements of the tridiagonal matrix .
4: – IntegerInput
On entry: , the number of eigenvectors to be returned.
Constraint:
.
5: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array w
must be at least
.
On entry: the eigenvalues of the tridiagonal matrix stored in to , as returned by f08jjf with . Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.
Constraint:
if , , for .
6: – Integer arrayInput
Note: the dimension of the array iblock
must be at least
.
On entry: the first elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by f08jjf with . If the eigenvalues were not computed by f08jjf with , set
to , for .
Constraint:
, for .
7: – Integer arrayInput
Note: the dimension of the array isplit
must be at least
.
On entry: the points at which breaks up into sub-matrices, as returned by f08jjf with . If the eigenvalues were not computed by f08jjf with , set to n.
8: – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z
must be at least
.
On exit: the eigenvectors, stored as columns of ; the th column corresponds to the th specified eigenvalue, unless (in which case see Section 6).
9: – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08jxf is called.
Constraint:
.
10: – Real (Kind=nag_wp) arrayWorkspace
11: – Integer arrayWorkspace
12: – Integer arrayOutput
On exit: if , the first elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to .
13: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
eigenvectors (as indicated by argument ifailv) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.
7Accuracy
Each computed eigenvector is the exact eigenvector of a nearby matrix , such that
where is the machine precision. Hence the residual is small:
However, a set of eigenvectors computed by this routine may not be orthogonal to so high a degree of accuracy as those computed by f08jsf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08jxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jxf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.