f08quf reorders the Schur factorization of a complex general matrix so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the Schur form. The routine also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.

2 Specification

Fortran Interface

Subroutine f08quf (

job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info)

Integer, Intent (In)	::	n, ldt, ldq, lwork
Integer, Intent (Out)	::	m, info
Real (Kind=nag_wp), Intent (Out)	::	s, sep
Complex (Kind=nag_wp), Intent (Inout)	::	t(ldt,), q(ldq,), w(*)
Complex (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Logical, Intent (In)	::	select(*)
Character (1), Intent (In)	::	job, compq

C Header Interface

#include <nag.h>

void

f08quf_ (const char *job, const char *compq, const logical sel[], const Integer *n, Complex t[], const Integer *ldt, Complex q[], const Integer *ldq, Complex w[], Integer *m, double *s, double *sep, Complex work[], const Integer *lwork, Integer *info, const Charlen length_job, const Charlen length_compq)

The routine may be called by the names f08quf, nagf_lapackeig_ztrsen or its LAPACK name ztrsen.

3 Description

f08quf reorders the Schur factorization of a complex general matrix

A = Q T Q^{H}

, so that a selected cluster of eigenvalues appears in the leading diagonal elements of the Schur form.

The reordered Schur form

\tilde{T}

is computed by a unitary similarity transformation:

\tilde{T} = Z^{H} T Z

. Optionally the updated matrix

\tilde{Q}

of Schur vectors is computed as

\tilde{Q} = Q Z

, giving

A = \tilde{Q} \tilde{T} {\tilde{Q}}^{H}

Let

\tilde{T} = (\begin{matrix} T_{11} & T_{12} \\ 0 & T_{22} \end{matrix})

, where the selected eigenvalues are precisely the eigenvalues of the leading

m \times m

sub-matrix

T_{11}

. Let

\tilde{Q}

be correspondingly partitioned as

(\begin{matrix} Q_{1} & Q_{2} \end{matrix})

where

Q_{1}

consists of the first

m

columns of

Q

. Then

A Q_{1} = Q_{1} T_{11}

, and so the

m

columns of

Q_{1}

form an orthonormal basis for the invariant subspace corresponding to the selected cluster of eigenvalues.

Optionally the routine also computes estimates of the reciprocal condition numbers of the average of the cluster of eigenvalues and of the invariant subspace.

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 condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.

$job ='N'$: No condition numbers are required.
$job ='E'$: Only the condition number for the cluster of eigenvalues is computed.
$job ='V'$: Only the condition number for the invariant subspace is computed.
$job ='B'$: Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.

Constraint:

job ='N'

'E'

'V'

'B'

2: $compq$ – Character(1) Input

On entry: indicates whether the matrix

Q

of Schur vectors is to be updated.

$compq ='V'$: The matrix $Q$ of Schur vectors is updated.
$compq ='N'$: No Schur vectors are updated.

Constraint:

compq ='V'

'N'

3: $select (*)$ – Logical array Input

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

\max (1, n)

On entry: specifies the eigenvalues in the selected cluster. To select a complex eigenvalue

λ_{j}

select (j)

must be set .TRUE..

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

\max (1, n)

On entry: the

n \times n

upper triangular matrix

T

, as returned by f08psf.

On exit: t is overwritten by the updated matrix

\tilde{T}

6: $ldt$ – Integer Input

On entry: the first dimension of the array t as declared in the (sub)program from which f08quf is called.

Constraint:

ldt \geq \max (1, n)

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

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

\max (1, n)

compq ='V'

and at least

1

compq ='N'

On entry: if

compq ='V'

, q must contain the

n \times n

unitary matrix

Q

of Schur vectors, as returned by f08psf.

On exit: if

compq ='V'

, q contains the updated matrix of Schur vectors; the first

m

columns of

Q

form an orthonormal basis for the specified invariant subspace.

compq ='N'

, q is not referenced.

8: $ldq$ – Integer Input

On entry: the first dimension of the array q as declared in the (sub)program from which f08quf is called.

Constraints:

if $compq ='V'$ , $ldq \geq \max (1, n)$ ;
if $compq ='N'$ , $ldq \geq 1$ .

9: $w (*)$ – Complex (Kind=nag_wp) array Output

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

\max (1, n)

On exit: the reordered eigenvalues of

\tilde{T}

. The eigenvalues are stored in the same order as on the diagonal of

\tilde{T}

10: $m$ – Integer Output

On exit:

m

, the dimension of the specified invariant subspace, which is the same as the number of selected eigenvalues (see select);

0 \leq m \leq n

11: $s$ – Real (Kind=nag_wp) Output

On exit: if

job ='E'

'B'

, s is a lower bound on the reciprocal condition number of the average of the selected cluster of eigenvalues. If

m = 0

n

s = 1

job ='N'

'V'

, s is not referenced.

12: $sep$ – Real (Kind=nag_wp) Output

On exit: if

job ='V'

'B'

, sep is the estimated reciprocal condition number of the specified invariant subspace. If

m = 0

n

sep = ‖ T ‖

job ='N'

'E'

, sep is not referenced.

13: $work (\max (1, lwork))$ – Complex (Kind=nag_wp) array Workspace

On exit: if

info = 0

, the real part of

work (1)

contains the minimum value of lwork required for optimal performance.

14: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08quf is called, unless

lwork = −1

, in which case a workspace query is assumed and the routine only calculates the minimum dimension of work.

Constraints:

if $job ='N'$ , $lwork \geq 1$ or $lwork = −1$ ;
if $job ='E'$ , $lwork \geq \max (1, m \times (n - m))$ or $lwork = −1$ ;
if $job ='V'$ or $'B'$ , $lwork \geq \max (1, 2 m \times (n - m))$ or $lwork = −1$ .

The actual amount of workspace required cannot exceed

n^{2} / 4

job ='E'

n^{2} / 2

job ='V'

'B'

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

The computed matrix

\tilde{T}

is similar to a matrix

(T + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ T ‖}_{2},

and

ε

is the machine precision.

s cannot underestimate the true reciprocal condition number by more than a factor of

\sqrt{\min (m, n - m)}

. sep may differ from the true value by

\sqrt{m (n - m)}

. The angle between the computed invariant subspace and the true subspace is

\frac{O (ε) {‖ A ‖}_{2}}{sep}

The values of the eigenvalues are never changed by the reordering.

8 Parallelism and Performance

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

f08quf 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.

9 Further Comments

The real analogue of this routine is f08qgf.

10 Example

This example reorders the Schur factorization of the matrix

A = Q T Q^{H}

such that the eigenvalues stored in elements

t_{11}

and

t_{44}

appear as the leading elements on the diagonal of the reordered matrix

\tilde{T}

, where

T = (\begin{array}{r} - 6.0004 - 6.9999 i & 0.3637 - 0.3656 i & - 0.1880 + 0.4787 i & 0.8785 - 0.2539 i \\ 0.0000 + 0.0000 i & - 5.0000 + 2.0060 i & - 0.0307 - 0.7217 i & - 0.2290 + 0.1313 i \\ 0.0000 + 0.0000 i & 0.0000 + 0.0000 i & 7.9982 - 0.9964 i & 0.9357 + 0.5359 i \\ 0.0000 + 0.0000 i & 0.0000 + 0.0000 i & 0.0000 + 0.0000 i & 3.0023 - 3.9998 i \end{array})

and

Q = (\begin{array}{r} - 0.8347 - 0.1364 i & - 0.0628 + 0.3806 i & 0.2765 - 0.0846 i & 0.0633 - 0.2199 i \\ 0.0664 - 0.2968 i & 0.2365 + 0.5240 i & - 0.5877 - 0.4208 i & 0.0835 + 0.2183 i \\ - 0.0362 - 0.3215 i & 0.3143 - 0.5473 i & 0.0576 - 0.5736 i & 0.0057 - 0.4058 i \\ 0.0086 + 0.2958 i & - 0.3416 - 0.0757 i & - 0.1900 - 0.1600 i & 0.8327 - 0.1868 i \end{array}) .

The original matrix

A

is given in f08ntf.

f08qu: FL CL CPP AD PY MB

NAG FL Interfacef08quf (ztrsen)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08quf (ztrsen)