Integer, Intent (In)	::	n, ldt, ldvl, ldvr, mm
Integer, Intent (Out)	::	m, info
Real (Kind=nag_wp), Intent (Out)	::	rwork(n)
Complex (Kind=nag_wp), Intent (Inout)	::	t(ldt,), vl(ldvl,), vr(ldvr,*)
Complex (Kind=nag_wp), Intent (Out)	::	work(2*n)
Logical, Intent (In)	::	select(*)
Character (1), Intent (In)	::	job, howmny

C Header Interface

#include <nag.h>

void

f08qxf_ (const char *job, const char *howmny, const logical sel[], const Integer *n, Complex t[], const Integer *ldt, Complex vl[], const Integer *ldvl, Complex vr[], const Integer *ldvr, const Integer *mm, Integer *m, Complex work[], double rwork[], Integer *info, const Charlen length_job, const Charlen length_howmny)

The routine may be called by the names f08qxf, nagf_lapackeig_ztrevc or its LAPACK name ztrevc.

3 Description

f08qxf computes left and/or right eigenvectors of a complex upper triangular matrix

T

. Such a matrix arises from the Schur factorization of a complex general matrix, as computed by f08psf, for example.

The right eigenvector

x

, and the left eigenvector

y

, corresponding to an eigenvalue

λ

, are defined by:

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

The routine can compute the eigenvectors corresponding to selected eigenvalues, or it can compute all the eigenvectors. In the latter case the eigenvectors may optionally be pre-multiplied by an input matrix

Q

. Normally

Q

is a unitary matrix from the Schur factorization of a matrix

A

A = Q T Q^{H}

; if

x

is a (left or right) eigenvector of

T

, then

Q x

is an eigenvector of

A

The eigenvectors are computed by forward or backward substitution. They are scaled so that

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

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: $howmny$ – Character(1) Input

On entry: indicates how many eigenvectors are to be computed.

$howmny ='A'$: All eigenvectors (as specified by job) are computed.
$howmny ='B'$: All eigenvectors (as specified by job) are computed and then pre-multiplied by the matrix $Q$ (which is overwritten).
$howmny ='S'$: Selected eigenvectors (as specified by job and select) are computed.

Constraint:

howmny ='A'

'B'

'S'

3: $select (*)$ – Logical array Input

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

n

howmny ='S'

, and at least

1

otherwise.

On entry: specifies which eigenvectors are to be computed if

howmny ='S'

. To obtain the eigenvector corresponding to the eigenvalue

λ_{j}

select (j)

must be set .TRUE..

howmny ='A'

'B'

, select is not referenced.

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

5: $t (ldt, *)$ – Complex (Kind=nag_wp) array Input/Output

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

n

On entry: the

n \times n

upper triangular matrix

T

, as returned by f08psf.

On exit: is used as internal workspace prior to being restored and hence is unchanged.

6: $ldt$ – Integer Input

On entry: the first dimension of the array t as declared in the (sub)program from which f08qxf is called.

Constraint:

ldt \geq \max (1, n)

7: $vl (ldvl, *)$ – Complex (Kind=nag_wp) array Input/Output

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

mm

job ='L'

'B'

On entry: if

howmny ='B'

and

job ='L'

'B'

, vl must contain an

n \times n

matrix

Q

(usually the matrix of Schur vectors returned by f08psf).

howmny ='A'

'S'

, vl need not be set.

On exit: if

job ='L'

'B'

, vl contains the computed left eigenvectors (as specified by howmny and 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.

8: $ldvl$ – Integer Input

On entry: the first dimension of the array vl as declared in the (sub)program from which f08qxf is called.

Constraints:

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

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

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

mm

job ='R'

'B'

On entry: if

howmny ='B'

and

job ='R'

'B'

, vr must contain an

n \times n

matrix

Q

(usually the matrix of Schur vectors returned by f08psf).

howmny ='A'

'S'

, vr need not be set.

On exit: if

job ='R'

'B'

, vr contains the computed right eigenvectors (as specified by howmny and 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.

10: $ldvr$ – Integer Input

On entry: the first dimension of the array vr as declared in the (sub)program from which f08qxf is called.

Constraints:

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

11: $mm$ – Integer Input

On entry: the number of columns in the arrays vl and/or vr. The precise number of columns required,

m

, is

n

howmny ='A'

'B'

; if

howmny ='S'

m

is the number of selected eigenvectors (see select), in which case

0 \leq m \leq n

Constraints:

if $howmny ='A'$ or $'B'$ , $mm \geq n$ ;
otherwise $mm \geq m$ .

12: $m$ – Integer Output

On exit:

m

, the number of selected eigenvectors. If

howmny ='A'

'B'

, m is set to

n

13: $work (2 \times n)$ – Complex (Kind=nag_wp) array Workspace

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

15: $info$ – 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.

7 Accuracy

x_{i}

is an exact right eigenvector, and

{\tilde{x}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{x}}_{i}, x_{i})

between them is bounded as follows:

θ ({\tilde{x}}_{i}, x_{i}) \leq \frac{c (n) ε {‖ T ‖}_{2}}{{sep}_{i}}

where

{sep}_{i}

is the reciprocal condition number of

x_{i}

The condition number

{sep}_{i}

may be computed by calling f08qyf.

8 Parallelism and Performance

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

f08qxf 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.