```      SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV,
\$                   INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          JOB, SIDE
INTEGER            IHI, ILO, INFO, LDV, M, N
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   SCALE( * ), V( LDV, * )
*     ..
*
*  Purpose
*  =======
*
*  DGEBAK forms the right or left eigenvectors of a real general matrix
*  by backward transformation on the computed eigenvectors of the
*  balanced matrix output by DGEBAL.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          Specifies the type of backward transformation required:
*          = 'N', do nothing, return immediately;
*          = 'P', do backward transformation for permutation only;
*          = 'S', do backward transformation for scaling only;
*          = 'B', do backward transformations for both permutation and
*                 scaling.
*          JOB must be the same as the argument JOB supplied to DGEBAL.
*
*  SIDE    (input) CHARACTER*1
*          = 'R':  V contains right eigenvectors;
*          = 'L':  V contains left eigenvectors.
*
*  N       (input) INTEGER
*          The number of rows of the matrix V.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) INTEGER
*          The integers ILO and IHI determined by DGEBAL.
*          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
*  SCALE   (input) DOUBLE PRECISION array, dimension (N)
*          Details of the permutation and scaling factors, as returned
*          by DGEBAL.
*
*  M       (input) INTEGER
*          The number of columns of the matrix V.  M >= 0.
*
*  V       (input/output) DOUBLE PRECISION array, dimension (LDV,M)
*          On entry, the matrix of right or left eigenvectors to be
*          transformed, as returned by DHSEIN or DTREVC.
*          On exit, V is overwritten by the transformed eigenvectors.
*
*  LDV     (input) INTEGER
*          The leading dimension of the array V. LDV >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ONE
PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            LEFTV, RIGHTV
INTEGER            I, II, K
DOUBLE PRECISION   S
*     ..
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           DSCAL, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Decode and Test the input parameters
*
RIGHTV = LSAME( SIDE, 'R' )
LEFTV = LSAME( SIDE, 'L' )
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
\$    .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -7
ELSE IF( LDV.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEBAK', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 )
\$   RETURN
IF( M.EQ.0 )
\$   RETURN
IF( LSAME( JOB, 'N' ) )
\$   RETURN
*
IF( ILO.EQ.IHI )
\$   GO TO 30
*
*     Backward balance
*
IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
*
IF( RIGHTV ) THEN
DO 10 I = ILO, IHI
S = SCALE( I )
CALL DSCAL( M, S, V( I, 1 ), LDV )
10       CONTINUE
END IF
*
IF( LEFTV ) THEN
DO 20 I = ILO, IHI
S = ONE / SCALE( I )
CALL DSCAL( M, S, V( I, 1 ), LDV )
20       CONTINUE
END IF
*
END IF
*
*     Backward permutation
*
*     For  I = ILO-1 step -1 until 1,
*              IHI+1 step 1 until N do --
*
30 CONTINUE
IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN
IF( RIGHTV ) THEN
DO 40 II = 1, N
I = II
IF( I.GE.ILO .AND. I.LE.IHI )
\$            GO TO 40
IF( I.LT.ILO )
\$            I = ILO - II
K = SCALE( I )
IF( K.EQ.I )
\$            GO TO 40
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
40       CONTINUE
END IF
*
IF( LEFTV ) THEN
DO 50 II = 1, N
I = II
IF( I.GE.ILO .AND. I.LE.IHI )
\$            GO TO 50
IF( I.LT.ILO )
\$            I = ILO - II
K = SCALE( I )
IF( K.EQ.I )
\$            GO TO 50
CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV )
50       CONTINUE
END IF
END IF
*
RETURN
*
*     End of DGEBAK
*
END

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