f08quc 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 function also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.

2 Specification

#include <nag.h>

void	f08quc (Nag_OrderType order, Nag_JobType job, Nag_ComputeQType compq, const Nag_Boolean select[], Integer n, Complex t[], Integer pdt, Complex q[], Integer pdq, Complex w[], Integer m, double s, double sep, NagError fail)

The function may be called by the names: f08quc, nag_lapackeig_ztrsen or nag_ztrsen.

3 Description

f08quc 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 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: specifies the eigenvalues in the selected cluster. To select a complex eigenvalue

λ_{j}

select [j - 1]

must be set Nag_TRUE.

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

6: $t [\dim]$ – Complex 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 triangular matrix

T

, as returned by f08psc.

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: $q [\dim]$ – Complex 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

unitary matrix

Q

of Schur vectors, as returned by f08psc.

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: $w [\dim]$ – Complex Output

Note: the dimension, dim, 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}

11: $m$ – 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

12: $s$ – 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

job = Nag_DoNothing

Nag_Subspace

, s is not referenced.

13: $sep$ – 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 ‖

job = Nag_DoNothing

Nag_EigVals

, sep is not referenced.

14: $fail$ – 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.

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.

f08quc 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 real analogue of this function is f08qgc.

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 f08ntc.

f08qu: FL CL CPP AD PY MB

NAG CL Interfacef08quc (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 CL Interface
f08quc (ztrsen)