f08ygc 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 function also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.

2 Specification

#include <nag.h>

void

f08ygc (Nag_OrderType order, Integer ijob, Nag_Boolean wantq, Nag_Boolean wantz, const Nag_Boolean select[], Integer n, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double q[], Integer pdq, double z[], Integer pdz, Integer *m, double *pl, double *pr, double dif[], NagError *fail)

The function may be called by the names: f08ygc, nag_lapackeig_dtgsen or nag_dtgsen.

3 Description

f08ygc 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 f08xcc, or f08xec with

job = Nag_Schur

. 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 function 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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $ijob$ – 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 (stored in $dif [0]$ and $dif [1]$ respectively).
$ijob = 3$: Estimate of ${Dif}_{u}$ and ${Dif}_{l}$ . $1$ -norm-based estimate (stored in $dif [0]$ and $dif [1]$ respectively). 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

3: $wantq$ – Nag_Boolean Input

On entry: if

wantq = Nag_TRUE

, update the left transformation matrix

Q

wantq = Nag_FALSE

, do not update

Q

4: $wantz$ – Nag_Boolean Input

On entry: if

wantz = Nag_TRUE

, update the right transformation matrix

Z

wantz = Nag_FALSE

, do not update

Z

5: $select [n]$ – const Nag_Boolean Input

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

λ_{j}

select [j - 1]

must be set to Nag_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 - 1]

select [j]

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

6: $n$ – Integer Input

On entry:

n

, the order of the matrices

S

and

T

Constraint:

n \geq 0

7: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the matrix

S

in the pair

(S, T)

On exit: the updated matrix

\hat{S}

8: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, n)

9: $b [\dim]$ – double Input/Output

Note: the dimension, dim, of the array b must be at least

\max (1, pdb \times n)

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the matrix

T

, in the pair

(S, T)

On exit: the updated matrix

\hat{T}

10: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraint:

pdb \geq \max (1, n)

11: $alphar [n]$ – double Output

On exit: see the description of beta.

12: $alphai [n]$ – double Output

On exit: see the description of beta.

13: $beta [n]$ – double Output

On exit:

alphar [j - 1] / beta [j - 1]

and

alphai [j - 1] / beta [j - 1]

are the real and imaginary parts respectively of the

j

th eigenvalue, for

j = 1, 2, \dots, n

alphai [j - 1]

is zero, then the

j

th eigenvalue is real; if positive then

alphai [j]

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.

14: $q [\dim]$ – double Input/Output

Note: the dimension, dim, of the array q must be at least

$\max (1, pdq \times n)$ when $wantq = Nag_TRUE$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Q

is stored in

$q [(j - 1) \times pdq + i - 1]$ when $order = Nag_ColMajor$ ;
$q [(i - 1) \times pdq + j - 1]$ when $order = Nag_RowMajor$ .

On entry: if

wantq = Nag_TRUE

, the

n \times n

matrix

Q

On exit: if

wantq = Nag_TRUE

, the updated matrix

Q \hat{Q}

wantq = Nag_FALSE

, q is not referenced.

15: $pdq$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array q.

Constraints:

if $wantq = Nag_TRUE$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

16: $z [\dim]$ – double Input/Output

Note: the dimension, dim, of the array z must be at least

$\max (1, pdz \times n)$ when $wantz = Nag_TRUE$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On entry: if

wantz = Nag_TRUE

, the

n \times n

matrix

Z

On exit: if

wantz = Nag_TRUE

, the updated matrix

Z \hat{Z}

wantz = Nag_FALSE

, z is not referenced.

17: $pdz$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

if $wantz = Nag_TRUE$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

18: $m$ – Integer * Output

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

19: $pl$ – double * Output

20: $pr$ – double * 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.

21: $dif [\dim]$ – double Output

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

2

On exit: if

ijob \geq 2

dif [0]

and

dif [1]

store the estimates of

{Dif}_{u}

and

{Dif}_{l}

ijob = 2

4

dif [0]

and

dif [1]

are

F

-norm-based upper bounds on

{Dif}_{u}

and

{Dif}_{l}

ijob = 3

5

dif [0]

and

dif [1]

are

1

-norm-based estimates of

{Dif}_{u}

and

{Dif}_{l}

m = 0

n

dif [0]

and

dif [1]

= {‖ (A, B) ‖}_{F}

ijob = 0

1

, dif is not referenced.

22: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONSTRAINT: Constraint: if $wantq = Nag_TRUE$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

Constraint: if $wantz = Nag_TRUE$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_INT: On entry, $ijob = ⟨ value ⟩$ .
Constraint: $0 \leq ijob \leq 5$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .

On entry, $pdq = ⟨ value ⟩$ .
Constraint: $pdq > 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SCHUR: 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 $dif [0]$ and $dif [1]$ , 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

f08ygc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this function is f08yuc.

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 CL Interfacef08ygc (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 CL Interface
f08ygc (dtgsen)