void	f08qtc (Nag_OrderType order, Nag_ComputeQType compq, Integer n, Complex t[], Integer pdt, Complex q[], Integer pdq, Integer ifst, Integer ilst, NagError *fail)

The function may be called by the names: f08qtc, nag_lapackeig_ztrexc or nag_ztrexc.

3 Description

f08qtc reorders the Schur factorization of a complex general matrix

A = Q T Q^{H}

, so that the diagonal element of

T

with row index ifst is moved to row ilst.

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}

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: $compq$ – 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

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

4: $t [\dim]$ – 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}

5: $pdt$ – 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)

6: $q [\dim]$ – 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.

On exit: if

compq = Nag_UpdateSchur

, q contains the updated matrix of Schur vectors.

compq = Nag_NotQ

, q is not referenced.

7: $pdq$ – 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 $compq = Nag_NotQ$ , $pdq \geq 1$ .

8: $ifst$ – Integer Input

9: $ilst$ – Integer Input

On entry: ifst and ilst must specify the reordering of the diagonal elements of

T

. The element with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements.

Constraint:

1 \leq ifst \leq n

and

1 \leq ilst \leq n

10: $fail$ – 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_INT_3: On entry, $n = ⟨ value ⟩$ , $ifst = ⟨ value ⟩$ and $ilst = ⟨ value ⟩$ .
Constraint: $1 \leq ifst \leq n$ and $1 \leq ilst \leq n$ .
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 exactly similar to a matrix

(T + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ T ‖}_{2},

and

ε

is the machine precision.

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.

f08qtc 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 total number of real floating-point operations is approximately

20 n r

compq = Nag_NotQ

, and

40 n r

compq = Nag_UpdateSchur

, where

r = | ifst - ilst |

The real analogue of this function is f08qfc.

10 Example

This example reorders the Schur factorization of the matrix

T

so that element

t_{11}

is moved to

t_{44}

, where

T = (\begin{array}{r} - 6.00 - 7.00 i & 0.36 - 0.36 i & - 0.19 + 0.48 i & 0.88 - 0.25 i \\ 0.00 + 0.00 i & - 5.00 + 2.00 i & - 0.03 - 0.72 i & - 0.23 + 0.13 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 8.00 - 1.00 i & 0.94 + 0.53 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 3.00 - 4.00 i \end{array}) .

f08qt: FL CL CPP AD PY MB

NAG CL Interfacef08qtc (ztrexc)

▸▿ 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
f08qtc (ztrexc)