F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08PXF (ZHSEIN)

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.

## 1  Purpose

F08PXF (ZHSEIN) computes selected left and/or right eigenvectors of a complex upper Hessenberg matrix corresponding to specified eigenvalues, by inverse iteration.

## 2  Specification

 SUBROUTINE F08PXF ( JOB, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO)
 INTEGER 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 $H$, corresponding to selected eigenvalues.
The right eigenvector $x$, and the left eigenvector $y$, corresponding to an eigenvalue $\lambda$, are defined by:
The eigenvectors are computed by inverse iteration. They are scaled so that $\mathrm{max}\phantom{\rule{0.25em}{0ex}}\left|\mathrm{Re}\left({x}_{i}\right)\right|+\left|\mathrm{Im}{x}_{i}\right|=1$.
If $H$ has been formed by reduction of a complex general matrix $A$ to upper Hessenberg form, then the eigenvectors of $H$ may be transformed to eigenvectors of $A$ 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:     JOB – CHARACTER(1)Input
On entry: indicates whether left and/or right eigenvectors are to be computed.
${\mathbf{JOB}}=\text{'R'}$
Only right eigenvectors are computed.
${\mathbf{JOB}}=\text{'L'}$
Only left eigenvectors are computed.
${\mathbf{JOB}}=\text{'B'}$
Both left and right eigenvectors are computed.
Constraint: ${\mathbf{JOB}}=\text{'R'}$, $\text{'L'}$ or $\text{'B'}$.
2:     EIGSRC – CHARACTER(1)Input
On entry: indicates whether the eigenvalues of $H$ (stored in W) were found using F08PSF (ZHSEQR).
${\mathbf{EIGSRC}}=\text{'Q'}$
The eigenvalues of $H$ were found using F08PSF (ZHSEQR); thus if $H$ has any zero subdiagonal elements (and so is block triangular), then the $j$th eigenvalue can be assumed to be an eigenvalue of the block containing the $j$th row/column. This property allows the routine to perform inverse iteration on just one diagonal block.
${\mathbf{EIGSRC}}=\text{'N'}$
No such assumption is made and the routine performs inverse iteration using the whole matrix.
Constraint: ${\mathbf{EIGSRC}}=\text{'Q'}$ or $\text{'N'}$.
3:     INITV – CHARACTER(1)Input
On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.
${\mathbf{INITV}}=\text{'N'}$
No initial estimates are supplied.
${\mathbf{INITV}}=\text{'U'}$
Initial estimates are supplied in VL and/or VR.
Constraint: ${\mathbf{INITV}}=\text{'N'}$ or $\text{'U'}$.
4:     SELECT($*$) – LOGICAL arrayInput
Note: the dimension of the array SELECT must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: specifies which eigenvectors are to be computed. To select the eigenvector corresponding to the eigenvalue ${\mathbf{W}}\left(j\right)$, ${\mathbf{SELECT}}\left(j\right)$ must be set to .TRUE..
5:     N – INTEGERInput
On entry: $n$, the order of the matrix $H$.
Constraint: ${\mathbf{N}}\ge 0$.
6:     H(LDH,$*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array H must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ upper Hessenberg matrix $H$.
7:     LDH – INTEGERInput
On entry: the first dimension of the array H as declared in the (sub)program from which F08PXF (ZHSEIN) is called.
Constraint: ${\mathbf{LDH}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     W($*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array W must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the eigenvalues of the matrix $H$. If ${\mathbf{EIGSRC}}=\text{'Q'}$, 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:     VL(LDVL,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array VL must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{MM}}\right)$ if ${\mathbf{JOB}}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{JOB}}=\text{'R'}$.
On entry: if ${\mathbf{INITV}}=\text{'U'}$ and ${\mathbf{JOB}}=\text{'L'}$ or $\text{'B'}$, 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).
If ${\mathbf{INITV}}=\text{'N'}$, VL need not be set.
On exit: if ${\mathbf{JOB}}=\text{'L'}$ or $\text{'B'}$, 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.
If ${\mathbf{JOB}}=\text{'R'}$, VL is not referenced.
10:   LDVL – INTEGERInput
On entry: the first dimension of the array VL as declared in the (sub)program from which F08PXF (ZHSEIN) is called.
Constraints:
• if ${\mathbf{JOB}}=\text{'L'}$ or $\text{'B'}$, ${\mathbf{LDVL}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• if ${\mathbf{JOB}}=\text{'R'}$, ${\mathbf{LDVL}}\ge 1$.
11:   VR(LDVR,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array VR must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{MM}}\right)$ if ${\mathbf{JOB}}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{JOB}}=\text{'L'}$.
On entry: if ${\mathbf{INITV}}=\text{'U'}$ and ${\mathbf{JOB}}=\text{'R'}$ or $\text{'B'}$, 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).
If ${\mathbf{INITV}}=\text{'N'}$, VR need not be set.
On exit: if ${\mathbf{JOB}}=\text{'R'}$ or $\text{'B'}$, 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.
If ${\mathbf{JOB}}=\text{'L'}$, VR is not referenced.
12:   LDVR – INTEGERInput
On entry: the first dimension of the array VR as declared in the (sub)program from which F08PXF (ZHSEIN) is called.
Constraints:
• if ${\mathbf{JOB}}=\text{'R'}$ or $\text{'B'}$, ${\mathbf{LDVR}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• if ${\mathbf{JOB}}=\text{'L'}$, ${\mathbf{LDVR}}\ge 1$.
13:   MM – INTEGERInput
On entry: the number of columns in the arrays VL and/or VR . The actual number of columns required, $\mathit{m}$, is obtained by counting $1$ for each selected real eigenvector and $2$ for each selected complex eigenvector (see SELECT); $0\le \mathit{m}\le n$.
Constraint: ${\mathbf{MM}}\ge \mathit{m}$.
14:   M – INTEGEROutput
On exit: $\mathit{m}$, the number of selected eigenvectors.
15:   WORK(${\mathbf{N}}×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
16:   RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
17:   IFAILL($*$) – INTEGER arrayOutput
Note: the dimension of the array IFAILL must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{MM}}\right)$ if ${\mathbf{JOB}}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{JOB}}=\text{'R'}$.
On exit: if ${\mathbf{JOB}}=\text{'L'}$ or $\text{'B'}$, then ${\mathbf{IFAILL}}\left(i\right)=0$ if the selected left eigenvector converged and ${\mathbf{IFAILL}}\left(i\right)=j>0$ if the eigenvector stored in the $i$th row or column of VL (corresponding to the $j$th eigenvalue) failed to converge.
If ${\mathbf{JOB}}=\text{'R'}$, IFAILL is not referenced.
18:   IFAILR($*$) – INTEGER arrayOutput
Note: the dimension of the array IFAILR must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{MM}}\right)$ if ${\mathbf{JOB}}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{JOB}}=\text{'L'}$.
On exit: if ${\mathbf{JOB}}=\text{'R'}$ or $\text{'B'}$, then ${\mathbf{IFAILR}}\left(i\right)=0$ if the selected right eigenvector converged and ${\mathbf{IFAILR}}\left(i\right)=j>0$ if the eigenvector stored in the $i$th column of VR (corresponding to the $j$th eigenvalue) failed to converge.
If ${\mathbf{JOB}}=\text{'L'}$, IFAILR is not referenced.
19:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, then $i$ 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 ${x}_{i}$ is the exact eigenvector of a nearby matrix $A+{E}_{i}$, such that $‖{E}_{i}‖=\mathit{O}\left(\epsilon \right)‖A‖$. Hence the residual is small:
 $Axi - λixi = Oε A .$
However, eigenvectors corresponding to close or coincident eigenvalues may not accurately span the relevant subspaces.
Similar remarks apply to computed left eigenvectors.

The real analogue of this routine is F08PKF (DHSEIN).

## 9  Example

See Section 9 in F08NUF (ZUNMHR).