naginterfaces.library.lapackeig.dgebak

naginterfaces.library.lapackeig.dgebak(job, side, ilo, ihi, scale, v)

dgebak transforms eigenvectors of a balanced matrix to those of the original real nonsymmetric matrix.

For full information please refer to the NAG Library document for f08nj

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f08/f08njf.html

Parameters

jobstr, length 1

This must be the same argument $job$ as supplied to dgebal().

sidestr, length 1

Indicates whether left or right eigenvectors are to be transformed.

$side=\text{'L'}$

The left eigenvectors are transformed.

$side=\text{'R'}$

The right eigenvectors are transformed.

iloint

The values $i_{lo}$ and $i_{hi}$, as returned by dgebal().

ihiint

The values $i_{lo}$ and $i_{hi}$, as returned by dgebal().

scalefloat, array-like, shape $\left(n\right)$

Details of the permutations and/or the scaling factors used to balance the original real nonsymmetric matrix, as returned by dgebal().

vfloat, array-like, shape $(n,m)$

The matrix of left or right eigenvectors to be transformed.

Returns

vfloat, ndarray, shape $(n,m)$

The transformed eigenvectors.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $job$.

Constraint: $job=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.

(errno $-2$)

On entry, error in parameter $side$.

Constraint: $side=\text{'L'}$ or $\text{'R'}$.

(errno $-3$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $ilo$.

(errno $-5$)

On entry, error in parameter $ihi$.

(errno $-7$)

On entry, error in parameter $\text{m}$.

Constraint: $m\ge 0$.

Notes

dgebak is intended to be used after a real nonsymmetric matrix $A$ has been balanced by dgebal(), and eigenvectors of the balanced matrix $A_{22}^{\prime \prime }$ have subsequently been computed.

For a description of balancing, see the document for dgebal(). The balanced matrix $A^{\prime \prime }$ is obtained as $A^{\prime \prime }=DPA{P}^T{D}^{-1}$, 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 $A^{\prime \prime }$, $P^T{D}^{-1}x$ is a right eigenvector of $A$;

if $y$ is a left eigenvector of $A^{\prime \prime }$, $P^{T}Dy$ is a left eigenvector of $A$.