Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
F08PXF (ZHSEIN) computes selected left and/or right eigenvectors of a complex upper Hessenberg matrix corresponding to specified eigenvalues, by inverse iteration.
N, LDH, LDVL, LDVR, MM, M, IFAILL(*), IFAILR(*), INFO
REAL (KIND=nag_wp)
RWORK(N)
COMPLEX (KIND=nag_wp)
H(LDH,*), W(*), VL(LDVL,*), VR(LDVR,*), WORK(N*N)
LOGICAL
SELECT(*)
CHARACTER(1)
JOB, EIGSRC, INITV
The routine may be called by its
LAPACK
name zhsein.
3 Description
F08PXF (ZHSEIN) computes left and/or right eigenvectors of a complex upper Hessenberg matrix , corresponding to selected eigenvalues.
The right eigenvector , and the left eigenvector , corresponding to an eigenvalue , are defined by:
The eigenvectors are computed by inverse iteration. They are scaled so that
.
If has been formed by reduction of a complex general matrix to upper Hessenberg form, then the eigenvectors of may be transformed to eigenvectors of by a call to F08NUF (ZUNMHR).
4 References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
1: – CHARACTER(1)Input
On entry: indicates whether left and/or right eigenvectors are to be computed.
Only right eigenvectors are computed.
Only left eigenvectors are computed.
Both left and right eigenvectors are computed.
Constraint:
, or .
2: – CHARACTER(1)Input
On entry: indicates whether the eigenvalues of (stored in W) were found using F08PSF (ZHSEQR).
The eigenvalues of were found using F08PSF (ZHSEQR); thus if has any zero subdiagonal elements (and so is block triangular), then the th eigenvalue can be assumed to be an eigenvalue of the block containing the th row/column. This property allows the routine to perform inverse iteration on just one diagonal block.
No such assumption is made and the routine performs inverse iteration using the whole matrix.
Constraint:
or .
3: – CHARACTER(1)Input
On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.
Note: the dimension of the array SELECT
must be at least
.
On entry: specifies which eigenvectors are to be computed. To select the eigenvector corresponding to the eigenvalue , must be set to .TRUE..
5: – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
6: – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array H
must be at least
.
On entry: the by upper Hessenberg matrix .
7: – INTEGERInput
On entry: the first dimension of the array H as declared in the (sub)program from which F08PXF (ZHSEIN) is called.
Constraint:
.
8: – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array W
must be at least
.
On entry: the eigenvalues of the matrix . If , the array must be exactly as returned by F08PSF (ZHSEQR).
On exit: the real parts of some elements of W may be modified, as close eigenvalues are perturbed slightly in searching for independent eigenvectors.
9: – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array VL
must be at least
if or and at least if .
On entry: if and or , VL must contain starting vectors for inverse iteration for the left eigenvectors. Each starting vector must be stored in the same column as will be used to store the corresponding eigenvector (see below).
On exit: if or , VL contains the computed left eigenvectors (as specified by SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.
On entry: the first dimension of the array VL as declared in the (sub)program from which F08PXF (ZHSEIN) is called.
Constraints:
if or , ;
if , .
11: – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array VR
must be at least
if or and at least if .
On entry: if and or , VR must contain starting vectors for inverse iteration for the right eigenvectors. Each starting vector must be stored in the same column as will be used to store the corresponding eigenvector (see below).
On exit: if or , VR contains the computed right eigenvectors (as specified by SELECT). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.
On entry: the first dimension of the array VR as declared in the (sub)program from which F08PXF (ZHSEIN) is called.
Constraints:
if or , ;
if , .
13: – INTEGERInput
On entry: the number of columns in the arrays VL and/or VR . The actual number of columns required, , is obtained by counting for each selected real eigenvector and for each selected complex eigenvector (see SELECT); .
Constraint:
.
14: – INTEGEROutput
On exit: , the number of selected eigenvectors.
15: – COMPLEX (KIND=nag_wp) arrayWorkspace
16: – REAL (KIND=nag_wp) arrayWorkspace
17: – INTEGER arrayOutput
Note: the dimension of the array IFAILL
must be at least
if or and at least if .
On exit: if or , then if the selected left eigenvector converged and if the eigenvector stored in the th row or column of VL (corresponding to the th eigenvalue) failed to converge.
Note: the dimension of the array IFAILR
must be at least
if or and at least if .
On exit: if or , then if the selected right eigenvector converged and if the eigenvector stored in the th column of VR (corresponding to the th eigenvalue) failed to converge.
On exit: unless the routine detects an error (see Section 6).
6 Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , then eigenvectors (as indicated by the parameters IFAILL and/or IFAILR above) failed to converge. The corresponding columns of VL and/or VR contain no useful information.
7 Accuracy
Each computed right eigenvector is the exact eigenvector of a nearby matrix , such that . Hence the residual is small:
However, eigenvectors corresponding to close or coincident eigenvalues may not accurately span the relevant subspaces.
Similar remarks apply to computed left eigenvectors.
8 Parallelism and Performance
F08PXF (ZHSEIN) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08PXF (ZHSEIN) 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.