NAG Library Routine Document

F08NJF (DGEBAK)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

1 Purpose

F08NJF (DGEBAK) transforms eigenvectors of a balanced matrix to those of the original real nonsymmetric matrix.

2 Specification

SUBROUTINE F08NJF (

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

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

The routine may be called by its LAPACK name dgebak.

3 Description

F08NJF (DGEBAK) is intended to be used after a real nonsymmetric matrix

A

has been balanced by F08NHF (DGEBAL), and eigenvectors of the balanced matrix

A_{22}^{''}

have subsequently been computed.

For a description of balancing, see the document for F08NHF (DGEBAL). 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 F08NHF (DGEBAL).

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 F08NHF (DGEBAL).

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 real nonsymmetric matrix, as returned by F08NHF (DGEBAL).

7: M – INTEGERInput

On entry:

m

, the number of columns of the matrix of eigenvectors.

Constraint:

M \geq 0

8: V(LDV, $*$ ) – REAL (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 F08NJF (DGEBAK) 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 Further Comments

The total number of floating point operations is approximately proportional to

n m

The complex analogue of this routine is F08NWF (ZGEBAK).

9 Example

See Section 9 in F08NHF (DGEBAL).

F08NJF (DGEBAK) (PDF version)

F08 Chapter Contents

F08 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentF08NJF (DGEBAK)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine Document

F08NJF (DGEBAK)