# ?gebak

Transforms eigenvectors of a balanced matrix to those of the original nonsymmetric matrix.

## Syntax

FORTRAN 77:

call sgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info)

call dgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info)

call cgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info)

call zgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info)

Fortran 95:

call gebak(v, scale [,ilo] [,ihi] [,job] [,side] [,info])

## Description

This routine is intended to be used after a matrix A has been balanced by a call to ?gebal, and eigenvectors of the balanced matrix A''22 have subsequently been computed. For a description of balancing, see ?gebal. The balanced matrix A'' is obtained as `A''= D*P*A*PT*inv(D)`, where P is a permutation matrix and D is a diagonal scaling matrix. This routine transforms the eigenvectors as follows:

if x is a right eigenvector of A'', then PT*`inv(D)*`x is a right eigenvector of A; if x is a left eigenvector of A'', then `PT*D*y` is a left eigenvector of A.

## Input Parameters

job

  CHARACTER*1. Must be 'N' or 'P' or 'S' or 'B'. The same parameter job as supplied to ?gebal.

side

  CHARACTER*1. Must be 'L' or 'R'.

  If `side = 'L'` , then left eigenvectors are transformed.

  If `side = 'R'`, then right eigenvectors are transformed.

n

  INTEGER. The number of rows of the matrix of eigenvectors (n ≥ 0).

ilo, ihi

  INTEGER. The values ilo and ihi, as returned by ?gebal. (If `n > 0`, then `1 ≤ ilo ≤ ihi ≤ n`;

  if `n = 0`, then `ilo = 1` and `ihi = 0`.)

scale

  REAL for single-precision flavors

  DOUBLE PRECISION for double-precision flavors

  Array, DIMENSION at least max(1, n).

  Contains details of the permutations and/or the scaling factors used to balance the original general matrix, as returned by ?gebal.

m

  INTEGER. The number of columns of the matrix of eigenvectors (`m ≥ 0`).

v

  REAL for sgebak

  DOUBLE PRECISION for dgebak

  COMPLEX for cgebak

  DOUBLE COMPLEX for zgebak.

  Arrays:

  v (ldv,*) contains the matrix of left or right eigenvectors to be transformed.

  The second dimension of v must be at least max(1, m).

ldv

  INTEGER. The first dimension of v; at least max(1, n).

## Output Parameters

v

  Overwritten by the transformed eigenvectors.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine gebak interface are the following:

v

  Holds the matrix V of size (n,m).

scale

  Holds the vector of length (n).

ilo

  Default value for this argument is `ilo = 1`.

ihi

  Default value for this argument is `ihi = n`.

job

  Must be 'B', 'S', 'P', or 'N'. The default value is 'B'.

side

  Must be 'L' or 'R'. The default value is 'L'.

## Application Notes

The errors in this routine are negligible.

The approximate number of floating-point operations is approximately proportional to `m*n`.

Parent topic: Nonsymmetric Eigenvalue Problems