f08qgc reorders the Schur factorization of a real general matrix so that a selected cluster of eigenvalues appears in the leading elements or blocks on the diagonal of the Schur form. The function also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.

2 Specification

#include <nag.h>

void	f08qgc (Nag_OrderType order, Nag_JobType job, Nag_ComputeQType compq, const Nag_Boolean select[], Integer n, double t[], Integer pdt, double q[], Integer pdq, double wr[], double wi[], Integer m, double s, double sep, NagError fail)

The function may be called by the names: f08qgc, nag_lapackeig_dtrsen or nag_dtrsen.

3 Description

f08qgc reorders the Schur factorization of a real general matrix

A = Q T Q^{T}

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

The reordered Schur form

\tilde{T}

is computed by an orthogonal similarity transformation:

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

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 function 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: $order$ – 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: $job$ – Nag_JobType Input

On entry: indicates whether condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.

$job = Nag_DoNothing$: No condition numbers are required.
$job = Nag_EigVals$: Only the condition number for the cluster of eigenvalues is computed.
$job = Nag_Subspace$: Only the condition number for the invariant subspace is computed.
$job = Nag_DoBoth$: Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.

Constraint:

job = Nag_DoNothing

Nag_EigVals

Nag_Subspace

Nag_DoBoth

3: $compq$ – Nag_ComputeQType Input

On entry: indicates whether the matrix

Q

of Schur vectors is to be updated.

$compq = Nag_UpdateSchur$: The matrix $Q$ of Schur vectors is updated.
$compq = Nag_NotQ$: No Schur vectors are updated.

Constraint:

compq = Nag_UpdateSchur

Nag_NotQ

4: $select [\dim]$ – const Nag_Boolean Input

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

\max (1, n)

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

λ_{j}

select [j - 1]

must be set Nag_TRUE. To select a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

(corresponding to a

2 \times 2

diagonal block),

select [j - 1]

and/or

select [j]

must be set to Nag_TRUE. A complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded. See also Section 9.

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

6: $t [\dim]$ – double Input/Output

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

\max (1, pdt \times n)

The

(i, j)

th element of the matrix

T

is stored in

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

On entry: the

n \times n

upper quasi-triangular matrix

T

in canonical Schur form, as returned by f08pec. See also Section 9.

On exit: t is overwritten by the updated matrix

\tilde{T}

7: $pdt$ – Integer Input

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

Constraint:

pdt \geq \max (1, n)

8: $q [\dim]$ – double Input/Output

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

$\max (1, pdq \times n)$ when $compq = Nag_UpdateSchur$ ;
$1$ when $compq = Nag_NotQ$ .

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

compq = Nag_UpdateSchur

, q must contain the

n \times n

orthogonal matrix

Q

of Schur vectors, as returned by f08pec.

On exit: if

compq = Nag_UpdateSchur

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

m

columns of

Q

form an orthonormal basis for the specified invariant subspace.

compq = Nag_NotQ

, q is not referenced.

9: $pdq$ – Integer Input

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

Constraints:

if $compq = Nag_UpdateSchur$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .

10: $wr [\dim]$ – double Output

11: $wi [\dim]$ – double Output

Note: the dimension, dim, of the arrays wr and wi must be at least

\max (1, n)

On exit: the real and imaginary parts, respectively, of the reordered eigenvalues of

\tilde{T}

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

\tilde{T}

; see Section 9 for details. Note that if a complex eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering.

12: $m$ – Integer * Output

On exit:

m

, the dimension of the specified invariant subspace. The value of

m

is obtained by counting

1

for each selected real eigenvalue and

2

for each selected complex conjugate pair of eigenvalues (see select);

0 \leq m \leq n

13: $s$ – double * Output

On exit: if

job = Nag_EigVals

Nag_DoBoth

, 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

; if

fail . code =

NE_REORDER (see Section 6), s is set to zero.

job = Nag_DoNothing

Nag_Subspace

, s is not referenced.

14: $sep$ – double * Output

On exit: if

job = Nag_Subspace

Nag_DoBoth

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

m = 0

n

sep = ‖ T ‖

; if

fail . code =

NE_REORDER (see Section 6), sep is set to zero.

job = Nag_DoNothing

Nag_EigVals

, sep is not referenced.

15: $fail$ – 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_ENUM_INT_2: On entry, $compq = ⟨ value ⟩$ , $pdq = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $compq = Nag_UpdateSchur$ , $pdq \geq \max (1, n)$ ;
if $compq = Nag_NotQ$ , $pdq \geq 1$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

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

On entry, $pdt = ⟨ value ⟩$ .
Constraint: $pdt > 0$ .
NE_INT_2: On entry, $pdt = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdt \geq \max (1, n)$ .
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_REORDER: The reordering of $T$ failed because a selected eigenvalue was too close to an unselected eigenvalue.
The reordering of $T$ failed because a selected eigenvalue was too close to an eigenvalue which was not selected; this error exit can only occur if at least one of the eigenvalues involved was complex. The problem is too ill-conditioned: consider modifying the selection of eigenvalues so that eigenvalues which are very close together are either all included in the cluster or all excluded. On exit, $T$ may have been partially reordered, but wr, wi and $Q$ (if requested) are updated consistently with $T$ ; s and sep (if requested) are both set to zero.

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}

Note that if a

2 \times 2

diagonal block is involved in the reordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a

2 \times 2

block to break into two

1 \times 1

blocks, i.e., for a pair of complex eigenvalues to become purely real. The values of real eigenvalues however are never changed by the reordering.

8 Parallelism and Performance

f08qgc 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 input matrix

T

must be in canonical Schur form, as is the output matrix

\tilde{T}

. This has the following structure.

If all the computed eigenvalues are real,

\tilde{T}

is upper triangular, and the diagonal elements of

\tilde{T}

are the eigenvalues;

wr [i - 1] = {\tilde{t}}_{i i}

, for

i = 1, 2, \dots, n

and

wi [i - 1] = 0.0

If some of the computed eigenvalues form complex conjugate pairs, then

\tilde{T}

has

2 \times 2

diagonal blocks. Each diagonal block has the form

(\begin{matrix} {\tilde{t}}_{i i} & {\tilde{t}}_{i, i + 1} \\ {\tilde{t}}_{i + 1, i} & {\tilde{t}}_{i + 1, i + 1} \end{matrix}) = (\begin{matrix} α & β \\ γ & α \end{matrix})

where

β γ < 0

. The corresponding eigenvalues are

α \pm \sqrt{β γ}

;

wr [i - 1] = wr [i] = α

;

wi [i - 1] = + \sqrt{| β γ |}

;

wi [i] = - wi [i - 1]

The complex analogue of this function is f08quc.

10 Example

This example reorders the Schur factorization of the matrix

A = Q T Q^{T}

such that the two real eigenvalues appear as the leading elements on the diagonal of the reordered matrix

\tilde{T}

, where

T = (\begin{array}{r} 0.7995 & - 0.1144 & 0.0060 & 0.0336 \\ 0.0000 & - 0.0994 & 0.2478 & 0.3474 \\ 0.0000 & - 0.6483 & - 0.0994 & 0.2026 \\ 0.0000 & 0.0000 & 0.0000 & - 0.1007 \end{array})

and

Q = (\begin{array}{r} 0.6551 & 0.1037 & 0.3450 & 0.6641 \\ 0.5236 & - 0.5807 & - 0.6141 & - 0.1068 \\ - 0.5362 & - 0.3073 & - 0.2935 & 0.7293 \\ 0.0956 & 0.7467 & - 0.6463 & 0.1249 \end{array}) .

The example program for f08qgc illustrates the computation of error bounds for the eigenvalues.

The original matrix

A

is given in f08nfc.

f08qg: FL CL CPP AD

NAG CL Interfacef08qgc (dtrsen)

▸▿ 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
f08qgc (dtrsen)