f08ygf reorders the generalized Schur factorization of a matrix pair in real generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements, or blocks on the diagonal of the generalized Schur form. The routine also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.

2 Specification

Fortran Interface

Subroutine f08ygf (

ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)

Integer, Intent (In)	::	ijob, n, lda, ldb, ldq, ldz, lwork, liwork
Integer, Intent (Out)	::	m, iwork(max(1,liwork)), info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), q(ldq,), z(ldz,), dif(*)
Real (Kind=nag_wp), Intent (Out)	::	alphar(n), alphai(n), beta(n), pl, pr, work(max(1,lwork))
Logical, Intent (In)	::	wantq, wantz, select(n)

C Header Interface

#include <nag.h>

void

f08ygf_ (const Integer *ijob, const logical *wantq, const logical *wantz, const logical sel[], const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, double alphar[], double alphai[], double beta[], double q[], const Integer *ldq, double z[], const Integer *ldz, Integer *m, double *pl, double *pr, double dif[], double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info)

The routine may be called by the names f08ygf, nagf_lapackeig_dtgsen or its LAPACK name dtgsen.

3 Description

f08ygf factorizes the generalized real

n \times n

matrix pair

(S, T)

in real generalized Schur form, using an orthogonal equivalence transformation as

S = \hat{Q} \hat{S} {\hat{Z}}^{T}, T = \hat{Q} \hat{T} {\hat{Z}}^{T},

where

(\hat{S}, \hat{T})

are also in real generalized Schur form and have the selected eigenvalues as the leading diagonal elements, or diagonal blocks. The leading columns of

Q

and

Z

are the generalized Schur vectors corresponding to the selected eigenvalues and form orthonormal subspaces for the left and right eigenspaces (deflating subspaces) of the pair

(S, T)

The pair

(S, T)

are in real generalized Schur form if

S

is block upper triangular with

1 \times 1

and

2 \times 2

diagonal blocks and

T

is upper triangular as returned, for example, by f08xcf, or f08xef with

job ='S'

. The diagonal elements, or blocks, define the generalized eigenvalues

(α_{i}, β_{i})

, for

i = 1, 2, \dots, n

, of the pair

(S, T)

. The eigenvalues are given by

λ_{i} = α_{i} / β_{i},

but are returned as the pair

(α_{i}, β_{i})

in order to avoid possible overflow in computing

λ_{i}

. Optionally, the routine returns reciprocals of condition number estimates for the selected eigenvalue cluster,

p

and

q

, the right and left projection norms, and of deflating subspaces,

{Dif}_{u}

and

{Dif}_{l}

. For more information see Sections 2.4.8 and 4.11 of Anderson et al. (1999).

S

and

T

are the result of a generalized Schur factorization of a matrix pair

(A, B)

A = Q S Z^{T}, B = Q T Z^{T}

then, optionally, the matrices

Q

and

Z

can be updated as

Q \hat{Q}

and

Z \hat{Z}

. Note that the condition numbers of the pair

(S, T)

are the same as those of the pair

(A, B)

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

5 Arguments

1: $ijob$ – Integer Input

On entry: specifies whether condition numbers are required for the cluster of eigenvalues (

p

and

q

) or the deflating subspaces (

{Dif}_{u}

and

{Dif}_{l}

$ijob = 0$: Only reorder with respect to select. No extras.
$ijob = 1$: Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster ( $p$ and $q$ ).
$ijob = 2$: The upper bounds on ${Dif}_{u}$ and ${Dif}_{l}$ . $F$ -norm-based estimate ( $dif (1 : 2)$ ).
$ijob = 3$: Estimate of ${Dif}_{u}$ and ${Dif}_{l}$ . $1$ -norm-based estimate ( $dif (1 : 2)$ ). About five times as expensive as $ijob = 2$ .
$ijob = 4$: Compute pl, pr and dif as in $ijob = 0$ , $1$ and $2$ . Economic version to get it all.
$ijob = 5$: Compute pl, pr and dif as in $ijob = 0$ , $1$ and $3$ .

Constraint:

0 \leq ijob \leq 5

2: $wantq$ – Logical Input

On entry: if

wantq = .TRUE.

, update the left transformation matrix

Q

wantq = .FALSE.

, do not update

Q

3: $wantz$ – Logical Input

On entry: if

wantz = .TRUE.

, update the right transformation matrix

Z

wantz = .FALSE.

, do not update

Z

4: $select (n)$ – Logical array Input

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

λ_{j}

select (j)

must be set to .TRUE..

To select a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

, corresponding to a

2 \times 2

diagonal block, either

select (j)

select (j + 1)

or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.

5: $n$ – Integer Input

On entry:

n

, the order of the matrices

S

and

T

Constraint:

n \geq 0

6: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the matrix

S

in the pair

(S, T)

On exit: the updated matrix

\hat{S}

7: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08ygf is called.

Constraint:

lda \geq \max (1, n)

8: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the matrix

T

, in the pair

(S, T)

On exit: the updated matrix

\hat{T}

9: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08ygf is called.

Constraint:

ldb \geq \max (1, n)

10: $alphar (n)$ – Real (Kind=nag_wp) array Output

On exit: see the description of beta.

11: $alphai (n)$ – Real (Kind=nag_wp) array Output

On exit: see the description of beta.

12: $beta (n)$ – Real (Kind=nag_wp) array Output

On exit:

alphar (j) / beta (j)

and

alphai (j) / beta (j)

are the real and imaginary parts respectively of the

j

th eigenvalue, for

j = 1, 2, \dots, n

alphai (j)

is zero, then the

j

th eigenvalue is real; if positive then

alphai (j + 1)

is negative, and the

j

th and

(j + 1)

st eigenvalues are a complex conjugate pair.

Conjugate pairs of eigenvalues correspond to the

2 \times 2

diagonal blocks of

\hat{S}

. These

2 \times 2

blocks can be reduced by applying complex unitary transformations to

(\hat{S}, \hat{T})

to obtain the complex Schur form

(\tilde{S}, \tilde{T})

, where

\tilde{S}

is triangular (and complex). In this form

alphar + i alphai

and beta are the diagonals of

\tilde{S}

and

\tilde{T}

respectively.

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

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

\max (1, n)

wantq = .TRUE.

, and at least

1

otherwise.

On entry: if

wantq = .TRUE.

, the

n \times n

matrix

Q

On exit: if

wantq = .TRUE.

, the updated matrix

Q \hat{Q}

wantq = .FALSE.

, q is not referenced.

14: $ldq$ – Integer Input

On entry: the first dimension of the array q as declared in the (sub)program from which f08ygf is called.

Constraints:

if $wantq = .TRUE.$ , $ldq \geq \max (1, n)$ ;
otherwise $ldq \geq 1$ .

15: $z (ldz, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

wantz = .TRUE.

, and at least

1

otherwise.

On entry: if

wantz = .TRUE.

, the

n \times n

matrix

Z

On exit: if

wantz = .TRUE.

, the updated matrix

Z \hat{Z}

wantz = .FALSE.

, z is not referenced.

16: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08ygf is called.

Constraints:

if $wantz = .TRUE.$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

17: $m$ – Integer Output

On exit: the dimension of the specified pair of left and right eigenspaces (deflating subspaces).

18: $pl$ – Real (Kind=nag_wp) Output

19: $pr$ – Real (Kind=nag_wp) Output

On exit: if

ijob = 1

4

5

, pl and pr are lower bounds on the reciprocal of the norm of ‘projections’

p

and

q

onto left and right eigenspaces with respect to the selected cluster.

0 < pl

pr \leq 1

m = 0

m = n

pl = pr = 1

ijob = 0

2

3

, pl and pr are not referenced.

20: $dif (*)$ – Real (Kind=nag_wp) array Output

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

2

On exit: if

ijob \geq 2

dif (1 : 2)

store the estimates of

{Dif}_{u}

and

{Dif}_{l}

ijob = 2

4

dif (1 : 2)

are

F

-norm-based upper bounds on

{Dif}_{u}

and

{Dif}_{l}

ijob = 3

5

dif (1 : 2)

are

1

-norm-based estimates of

{Dif}_{u}

and

{Dif}_{l}

m = 0

n

dif (1 : 2)

= {‖ (A, B) ‖}_{F}

ijob = 0

1

, dif is not referenced.

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

On exit: if

info = 0

work (1)

returns the minimum lwork.

ijob = 0

, work is not referenced.

22: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08ygf is called.

lwork = −1

, a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.

Constraints:

lwork \neq −1

if $n = 0$ , $lwork \geq 1$ ;
if $ijob = 1$ , $2$ or $4$ , $lwork \geq \max (4 \times n + 16, 2 \times m \times (n - m))$ ;
if $ijob = 3$ or $5$ , $lwork \geq \max (4 \times n + 16, 4 \times m \times (n - m))$ ;
otherwise $lwork \geq 4 \times n + 16$ .

23: $iwork (\max (1, liwork))$ – Integer array Workspace

On exit: if

info = 0

iwork (1)

returns the minimum liwork.

ijob = 0

, iwork is not referenced.

24: $liwork$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08ygf is called.

liwork = −1

Constraints:

liwork \neq −1

if $ijob = 1$ , $2$ or $4$ , $liwork \geq n + 6$ ;
if $ijob = 3$ or $5$ , $liwork \geq \max (2 \times m \times (n - m), n + 6)$ ;
otherwise $liwork \geq 1$ .

25: $info$ – 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.

$info = 1$: Reordering of $(S, T)$ failed because the transformed matrix pair would be too far from generalized Schur form; the problem is very ill-conditioned. $(S, T)$ may have been partially reordered. If requested, $0$ is returned in $dif (1)$ and $dif (2)$ , pl and pr.

7 Accuracy

The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices

(S + E)

and

(T + F)

, where

{‖ E ‖}_{2} = O ε {‖ S ‖}_{2} and {‖ F ‖}_{2} = O ε {‖ T ‖}_{2},

and

ε

is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem, and for information on the condition numbers returned.

8 Parallelism and Performance

f08ygf 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 complex analogue of this routine is f08yuf.

10 Example

This example reorders the generalized Schur factors

S

and

T

and update the matrices

Q

and

Z

given by

S = (\begin{array}{r} 4.0 & 1.0 & 1.0 & 2.0 \\ 0 & 3.0 & 4.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 6.0 \end{array}), T = (\begin{array}{r} 2.0 & 1.0 & 1.0 & 3.0 \\ 0 & 1.0 & 2.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 2.0 \end{array}),

Q = (\begin{array}{r} 1.0 & 0 & 0 & 0 \\ 0 & 1.0 & 0 & 0 \\ 0 & 0 & 1.0 & 0 \\ 0 & 0 & 0 & 1.0 \end{array}) and Z = (\begin{array}{r} 1.0 & 0 & 0 & 0 \\ 0 & 1.0 & 0 & 0 \\ 0 & 0 & 1.0 & 0 \\ 0 & 0 & 0 & 1.0 \end{array}),

selecting the first and fourth generalized eigenvalues to be moved to the leading positions. Bases for the left and right deflating subspaces, and estimates of the condition numbers for the eigenvalues and Frobenius norm based bounds on the condition numbers for the deflating subspaces are also output.

f08yg: FL CL CPP AD

NAG FL Interfacef08ygf (dtgsen)

▸▿ 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
f08ygf (dtgsen)