f08pxf computes selected left and/or right eigenvectors of a complex upper Hessenberg matrix corresponding to specified eigenvalues, by inverse iteration.

2 Specification

Fortran Interface

Subroutine f08pxf (

job, eigsrc, initv, select, n, h, ldh, w, vl, ldvl, vr, ldvr, mm, m, work, rwork, ifaill, ifailr, info)

Integer, Intent (In)	::	n, ldh, ldvl, ldvr, mm
Integer, Intent (Inout)	::	ifaill(), ifailr()
Integer, Intent (Out)	::	m, info
Real (Kind=nag_wp), Intent (Out)	::	rwork(n)
Complex (Kind=nag_wp), Intent (In)	::	h(ldh,*)
Complex (Kind=nag_wp), Intent (Inout)	::	w(), vl(ldvl,), vr(ldvr,*)
Complex (Kind=nag_wp), Intent (Out)	::	work(n*n)
Logical, Intent (In)	::	select(*)
Character (1), Intent (In)	::	job, eigsrc, initv

C Header Interface

#include <nag.h>

void

f08pxf_ (const char *job, const char *eigsrc, const char *initv, const logical sel[], const Integer *n, const Complex h[], const Integer *ldh, Complex w[], Complex vl[], const Integer *ldvl, Complex vr[], const Integer *ldvr, const Integer *mm, Integer *m, Complex work[], double rwork[], Integer ifaill[], Integer ifailr[], Integer *info, const Charlen length_job, const Charlen length_eigsrc, const Charlen length_initv)

The routine may be called by the names f08pxf, nagf_lapackeig_zhsein or its LAPACK name zhsein.

3 Description

f08pxf 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

λ

, are defined by:

H x = λ x and y^{H} H = λ y^{H} (or H^{H} y = \bar{λ} y) .

The eigenvectors are computed by inverse iteration. They are scaled so that

\max | Re (x_{i}) | + | Im x_{i} | = 1

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.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: $job$ – Character(1) Input

On entry: indicates whether left and/or right eigenvectors are to be computed.

$job ='R'$: Only right eigenvectors are computed.
$job ='L'$: Only left eigenvectors are computed.
$job ='B'$: Both left and right eigenvectors are computed.

Constraint:

job ='R'

'L'

'B'

2: $eigsrc$ – Character(1) Input

On entry: indicates whether the eigenvalues of

H

(stored in w) were found using f08psf.

$eigsrc ='Q'$: The eigenvalues of $H$ were found using f08psf; 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.
$eigsrc ='N'$: No such assumption is made and the routine performs inverse iteration using the whole matrix.

Constraint:

eigsrc ='Q'

'N'

3: $initv$ – Character(1) Input

On entry: indicates whether you are supplying initial estimates for the selected eigenvectors.

$initv ='N'$: No initial estimates are supplied.
$initv ='U'$: Initial estimates are supplied in vl and/or vr.

Constraint:

initv ='N'

'U'

4: $select (*)$ – Logical array Input

Note: the dimension of the array select must be at least

\max (1, n)

On entry: specifies which eigenvectors are to be computed. To select the eigenvector corresponding to the eigenvalue

w (j)

select (j)

must be set to .TRUE..

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

H

Constraint:

n \geq 0

6: $h (ldh, *)$ – Complex (Kind=nag_wp) array Input

Note: the second dimension of the array h must be at least

\max (1, n)

On entry: the

n \times n

upper Hessenberg matrix

H

. If a NaN is detected in h, the routine will return with

info = - 6

Constraint: No element of h is equal to NaN.

7: $ldh$ – Integer Input

On entry: the first dimension of the array h as declared in the (sub)program from which f08pxf is called.

Constraint:

ldh \geq \max (1, n)

8: $w (*)$ – Complex (Kind=nag_wp) array Input/Output

Note: the dimension of the array w must be at least

\max (1, n)

On entry: the eigenvalues of the matrix

H

. If

eigsrc ='Q'

, the array must be exactly as returned by f08psf.

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) array Input/Output

Note: the second dimension of the array vl must be at least

\max (1, mm)

job ='L'

'B'

and at least

1

job ='R'

On entry: if

initv ='U'

and

job ='L'

'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).

initv ='N'

, vl need not be set.

On exit: if

job ='L'

'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.

job ='R'

, vl is not referenced.

10: $ldvl$ – Integer Input

On entry: the first dimension of the array vl as declared in the (sub)program from which f08pxf is called.

Constraints:

if $job ='L'$ or $'B'$ , $ldvl \geq n$ ;
if $job ='R'$ , $ldvl \geq 1$ .

11: $vr (ldvr, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array vr must be at least

\max (1, mm)

job ='R'

'B'

and at least

1

job ='L'

On entry: if

initv ='U'

and

job ='R'

'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).

initv ='N'

, vr need not be set.

On exit: if

job ='R'

'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.

job ='L'

, vr is not referenced.

12: $ldvr$ – Integer Input

On entry: the first dimension of the array vr as declared in the (sub)program from which f08pxf is called.

Constraints:

if $job ='R'$ or $'B'$ , $ldvr \geq n$ ;
if $job ='L'$ , $ldvr \geq 1$ .

13: $mm$ – Integer Input

On entry: the number of columns in the arrays vl and/or vr. This must be an upper bound on the actual number of columns required, that is, the number of elements of select, in the first n, that are set to .TRUE..

Constraint:

mm \geq m

14: $m$ – Integer Output

On exit:

m

, the number of selected eigenvectors.

15: $work (n \times n)$ – Complex (Kind=nag_wp) array Workspace

16: $rwork (n)$ – Real (Kind=nag_wp) array Workspace

17: $ifaill (*)$ – Integer array Output

Note: the dimension of the array ifaill must be at least

\max (1, mm)

job ='L'

'B'

and at least

1

job ='R'

On exit: if

job ='L'

'B'

, then

ifaill (i) = 0

if the selected left eigenvector converged and

ifaill (i) = j > 0

if the eigenvector stored in the

i

th row or column of vl (corresponding to the

j

th eigenvalue) failed to converge.

job ='R'

, ifaill is not referenced.

18: $ifailr (*)$ – Integer array Output

Note: the dimension of the array ifailr must be at least

\max (1, mm)

job ='R'

'B'

and at least

1

job ='L'

On exit: if

job ='R'

'B'

, then

ifailr (i) = 0

if the selected right eigenvector converged and

ifailr (i) = j > 0

if the eigenvector stored in the

i

th column of vr (corresponding to the

j

th eigenvalue) failed to converge.

job ='L'

, ifailr is not referenced.

19: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: $⟨ value ⟩$ eigenvectors (as indicated by arguments ifaill and/or ifailr) 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} ‖ = O (ε) ‖ A ‖

. Hence the residual is small:

‖ A x_{i} - λ_{i} x_{i} ‖ = O (ε) ‖ A ‖ .

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

Background information to multithreading can be found in the Multithreading documentation.

f08pxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08pxf 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.