NAG Library Routine Document

F08NWF (ZGEBAK)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

1 Purpose

F08NWF (ZGEBAK) transforms eigenvectors of a balanced matrix to those of the original complex general matrix.

2 Specification

SUBROUTINE F08NWF (

JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)

INTEGER	N, ILO, IHI, M, LDV, INFO
REAL (KIND=nag_wp)	SCALE(*)
COMPLEX (KIND=nag_wp)	V(LDV,*)
CHARACTER(1)	JOB, SIDE

The routine may be called by its LAPACK name zgebak.

3 Description

F08NWF (ZGEBAK) is intended to be used after a complex general matrix

A

has been balanced by F08NVF (ZGEBAL), and eigenvectors of the balanced matrix

A_{22}^{''}

have subsequently been computed.

For a description of balancing, see the document for F08NVF (ZGEBAL). 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 routine 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 Parameters

1: $JOB$ – CHARACTER(1)Input

On entry: this must be the same parameter JOB as supplied to F08NVF (ZGEBAL).

Constraint:

JOB ='N'

'P'

'S'

'B'

2: $SIDE$ – CHARACTER(1)Input

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

$SIDE ='L'$: The left eigenvectors are transformed.
$SIDE ='R'$: The right eigenvectors are transformed.

Constraint:

SIDE ='L'

'R'

3: $N$ – INTEGERInput

On entry:

n

, the number of rows of the matrix of eigenvectors.

Constraint:

N \geq 0

4: $ILO$ – INTEGERInput

5: $IHI$ – INTEGERInput

On entry: the values

i_{lo}

and

i_{hi}

, as returned by F08NVF (ZGEBAL).

Constraints:

if $N > 0$ , $1 \leq ILO \leq IHI \leq N$ ;
if $N = 0$ , $ILO = 1$ and $IHI = 0$ .

6: $SCALE (*)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension 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 F08NVF (ZGEBAL).

7: $M$ – INTEGERInput

On entry:

m

, the number of columns of the matrix of eigenvectors.

Constraint:

M \geq 0

8: $V (LDV *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array V must be at least

\max (1, M)

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

On exit: the transformed eigenvectors.

9: $LDV$ – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F08NWF (ZGEBAK) is called.

Constraint:

LDV \geq \max (1, N)

10: $INFO$ – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$INFO < 0$: If $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The errors are negligible.

8 Parallelism and Performance

F08NWF (ZGEBAK) is not threaded by NAG in any implementation.

F08NWF (ZGEBAK) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.