NAG Library Function Document

nag_dgebak (f08njc)

+− 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

1 Purpose

nag_dgebak (f08njc) transforms eigenvectors of a balanced matrix to those of the original real nonsymmetric matrix.

2 Specification

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_dgebak (f08njc) is intended to be used after a real nonsymmetric matrix

A

has been balanced by nag_dgebal (f08nhc), and eigenvectors of the balanced matrix

A_{22}^{''}

have subsequently been computed.

For a description of balancing, see the document for nag_dgebal (f08nhc). 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_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: job – Nag_JobTypeInput

On entry: this must be the same argument job as supplied to nag_dgebal (f08nhc).

Constraint:

job = Nag_DoNothing

Nag_Permute

Nag_Scale

Nag_DoBoth

3: side – Nag_SideTypeInput

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 – IntegerInput

On entry:

n

, the number of rows of the matrix of eigenvectors.

Constraint:

n \geq 0

5: ilo – IntegerInput

6: ihi – IntegerInput

On entry: the values

i_{lo}

and

i_{hi}

, as returned by nag_dgebal (f08nhc).

Constraints:

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

7: scale[ $\dim$ ] – const doubleInput

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 real nonsymmetric matrix, as returned by nag_dgebal (f08nhc).

8: m – IntegerInput

On entry:

m

, the number of columns of the matrix of eigenvectors.

Constraint:

m \geq 0

9: v[ $\dim$ ] – doubleInput/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 – IntegerInput

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 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_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.

7 Accuracy

The errors are negligible.

8 Parallelism and Performance

nag_dgebak (f08njc) is not threaded by NAG in any implementation.

nag_dgebak (f08njc) 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.