NAG Library Function Document

nag_ztrexc (f08qtc)

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

3 Description

nag_ztrexc (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_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: compq – Nag_ComputeQTypeInput

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 – IntegerInput

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

4: t[ $\dim$ ] – ComplexInput/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

n

upper triangular matrix

T

, as returned by nag_zhseqr (f08psc).

On exit: t is overwritten by the updated matrix

\tilde{T}

5: pdt – IntegerInput

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[ $\dim$ ] – ComplexInput/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

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 – IntegerInput

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 – IntegerInput

9: ilst – IntegerInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

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

nag_ztrexc (f08qtc) is not threaded by NAG in any implementation.

nag_ztrexc (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 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 nag_dtrexc (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}) .

NAG Library Function Documentnag_ztrexc (f08qtc)

+− 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 Library Function Document

nag_ztrexc (f08qtc)