void	f08nwc (Nag_OrderType order, Nag_JobType job, Nag_SideType side, Integer n, Integer ilo, Integer ihi, const double scale[], Integer m, Complex v[], Integer pdv, NagError *fail)

The function may be called by the names: f08nwc, nag_lapackeig_zgebak or nag_zgebak.

3 Description

f08nwc is intended to be used after a complex general matrix

A

has been balanced by f08nvc, and eigenvectors of the balanced matrix

A_{22}^{''}

have subsequently been computed.

For a description of balancing, see the document for f08nvc. The balanced matrix

A^{''}

is obtained as

A^{''} = D P A P^{T} D^{- 1}

, where

P

is a permutation matrix and

D

is a diagonal scaling matrix. This function transforms left or right eigenvectors as follows:

if $x$ is a right eigenvector of $A^{''}$ , $P^{T} D^{- 1} x$ is a right eigenvector of $A$ ;
if $y$ is a left eigenvector of $A^{''}$ , $P^{T} D y$ is a left eigenvector of $A$ .

4 References

None.

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: this must be the same argument job as supplied to f08nvc.

Constraint:

job = Nag_DoNothing

Nag_Permute

Nag_Scale

Nag_DoBoth

3: $side$ – Nag_SideType Input

On entry: indicates whether left or right eigenvectors are to be transformed.

$side = Nag_LeftSide$: The left eigenvectors are transformed.
$side = Nag_RightSide$: The right eigenvectors are transformed.

Constraint:

side = Nag_LeftSide

Nag_RightSide

4: $n$ – Integer Input

On entry:

n

, the number of rows of the matrix of eigenvectors.

Constraint:

n \geq 0

5: $ilo$ – Integer Input

6: $ihi$ – Integer Input

On entry: the values

i_{lo}

and

i_{hi}

, as returned by f08nvc.

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

7: $scale [\dim]$ – const double Input

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

\max (1, n)

On entry: details of the permutations and/or the scaling factors used to balance the original complex general matrix, as returned by f08nvc.

8: $m$ – Integer Input

On entry:

m

, the number of columns of the matrix of eigenvectors.

Constraint:

m \geq 0

9: $v [\dim]$ – Complex Input/Output

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

$\max (1, pdv \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdv)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

V

is stored in

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

On entry: the matrix of left or right eigenvectors to be transformed.

On exit: the transformed eigenvectors.

10: $pdv$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ , $pdv \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdv \geq \max (1, m)$ .

11: $fail$ – 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_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

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

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

On entry, $pdv = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdv \geq \max (1, n)$ .
NE_INT_3: On entry, $n = ⟨ value ⟩$ , $ilo = ⟨ value ⟩$ and $ihi = ⟨ value ⟩$ .
Constraint: if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .
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 errors are negligible.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08nwc 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.