OpenBLAS/lapack-netlib/SRC/dtrevc3.f

*> \brief \b DTREVC3
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
*                           VR, LDVR, MM, M, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, SIDE
*       INTEGER            INFO, LDT, LDVL, LDVR, LWORK, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTREVC3 computes some or all of the right and/or left eigenvectors of
*> a real upper quasi-triangular matrix T.
*> Matrices of this type are produced by the Schur factorization of
*> a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
*>
*> The right eigenvector x and the left eigenvector y of T corresponding
*> to an eigenvalue w are defined by:
*>
*>    T*x = w*x,     (y**T)*T = w*(y**T)
*>
*> where y**T denotes the transpose of the vector y.
*> The eigenvalues are not input to this routine, but are read directly
*> from the diagonal blocks of T.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
*> input matrix. If Q is the orthogonal factor that reduces a matrix
*> A to Schur form T, then Q*X and Q*Y are the matrices of right and
*> left eigenvectors of A.
*>
*> This uses a Level 3 BLAS version of the back transformation.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R':  compute right eigenvectors only;
*>          = 'L':  compute left eigenvectors only;
*>          = 'B':  compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A':  compute all right and/or left eigenvectors;
*>          = 'B':  compute all right and/or left eigenvectors,
*>                  backtransformed by the matrices in VR and/or VL;
*>          = 'S':  compute selected right and/or left eigenvectors,
*>                  as indicated by the logical array SELECT.
*> \endverbatim
*>
*> \param[in,out] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
*>          computed.
*>          If w(j) is a real eigenvalue, the corresponding real
*>          eigenvector is computed if SELECT(j) is .TRUE..
*>          If w(j) and w(j+1) are the real and imaginary parts of a
*>          complex eigenvalue, the corresponding complex eigenvector is
*>          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
*>          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
*>          .FALSE..
*>          Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix T. N >= 0.
*> \endverbatim
*>
*> \param[in] T
*> \verbatim
*>          T is DOUBLE PRECISION array, dimension (LDT,N)
*>          The upper quasi-triangular matrix T in Schur canonical form.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of Schur vectors returned by DHSEQR).
*>          On exit, if SIDE = 'L' or 'B', VL contains:
*>          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
*>          if HOWMNY = 'B', the matrix Q*Y;
*>          if HOWMNY = 'S', the left eigenvectors of T specified by
*>                           SELECT, stored consecutively in the columns
*>                           of VL, in the same order as their
*>                           eigenvalues.
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part, and the second the imaginary part.
*>          Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.
*>          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of Schur vectors returned by DHSEQR).
*>          On exit, if SIDE = 'R' or 'B', VR contains:
*>          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
*>          if HOWMNY = 'B', the matrix Q*X;
*>          if HOWMNY = 'S', the right eigenvectors of T specified by
*>                           SELECT, stored consecutively in the columns
*>                           of VR, in the same order as their
*>                           eigenvalues.
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part and the second the imaginary part.
*>          Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.
*>          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR actually
*>          used to store the eigenvectors.
*>          If HOWMNY = 'A' or 'B', M is set to N.
*>          Each selected real eigenvector occupies one column and each
*>          selected complex eigenvector occupies two columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of array WORK. LWORK >= max(1,3*N).
*>          For optimum performance, LWORK >= N + 2*N*NB, where NB is
*>          the optimal blocksize.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2017
*
*  @precisions fortran d -> s
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The algorithm used in this program is basically backward (forward)
*>  substitution, with scaling to make the the code robust against
*>  possible overflow.
*>
*>  Each eigenvector is normalized so that the element of largest
*>  magnitude has magnitude 1; here the magnitude of a complex number
*>  (x,y) is taken to be |x| + |y|.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
     $                    VR, LDVR, MM, M, WORK, LWORK, INFO )
      IMPLICIT NONE
*
*  -- LAPACK computational routine (version 3.8.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2017
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, SIDE
      INTEGER            INFO, LDT, LDVL, LDVR, LWORK, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      DOUBLE PRECISION   T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      INTEGER            NBMIN, NBMAX
      PARAMETER          ( NBMIN = 8, NBMAX = 128 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLV, BOTHV, LEFTV, LQUERY, OVER, PAIR,
     $                   RIGHTV, SOMEV
      INTEGER            I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI,
     $                   IV, MAXWRK, NB, KI2
      DOUBLE PRECISION   BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
     $                   SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
     $                   XNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, ILAENV
      DOUBLE PRECISION   DDOT, DLAMCH
      EXTERNAL           LSAME, IDAMAX, ILAENV, DDOT, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DAXPY, DCOPY, DGEMV, DLALN2, DSCAL, XERBLA,
     $                   DGEMM, DLASET, DLABAD, DLACPY
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   X( 2, 2 )
      INTEGER            ISCOMPLEX( NBMAX )
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      BOTHV  = LSAME( SIDE, 'B' )
      RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
      LEFTV  = LSAME( SIDE, 'L' ) .OR. BOTHV
*
      ALLV  = LSAME( HOWMNY, 'A' )
      OVER  = LSAME( HOWMNY, 'B' )
      SOMEV = LSAME( HOWMNY, 'S' )
*
      INFO = 0
      NB = ILAENV( 1, 'DTREVC', SIDE // HOWMNY, N, -1, -1, -1 )
      MAXWRK = N + 2*N*NB
      WORK(1) = MAXWRK
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
         INFO = -1
      ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
         INFO = -8
      ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
         INFO = -10
      ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN
         INFO = -14
      ELSE
*
*        Set M to the number of columns required to store the selected
*        eigenvectors, standardize the array SELECT if necessary, and
*        test MM.
*
         IF( SOMEV ) THEN
            M = 0
            PAIR = .FALSE.
            DO 10 J = 1, N
               IF( PAIR ) THEN
                  PAIR = .FALSE.
                  SELECT( J ) = .FALSE.
               ELSE
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).EQ.ZERO ) THEN
                        IF( SELECT( J ) )
     $                     M = M + 1
                     ELSE
                        PAIR = .TRUE.
                        IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN
                           SELECT( J ) = .TRUE.
                           M = M + 2
                        END IF
                     END IF
                  ELSE
                     IF( SELECT( N ) )
     $                  M = M + 1
                  END IF
               END IF
   10       CONTINUE
         ELSE
            M = N
         END IF
*
         IF( MM.LT.M ) THEN
            INFO = -11
         END IF
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTREVC3', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Use blocked version of back-transformation if sufficient workspace.
*     Zero-out the workspace to avoid potential NaN propagation.
*
      IF( OVER .AND. LWORK .GE. N + 2*N*NBMIN ) THEN
         NB = (LWORK - N) / (2*N)
         NB = MIN( NB, NBMAX )
         CALL DLASET( 'F', N, 1+2*NB, ZERO, ZERO, WORK, N )
      ELSE
         NB = 1
      END IF
*
*     Set the constants to control overflow.
*
      UNFL = DLAMCH( 'Safe minimum' )
      OVFL = ONE / UNFL
      CALL DLABAD( UNFL, OVFL )
      ULP = DLAMCH( 'Precision' )
      SMLNUM = UNFL*( N / ULP )
      BIGNUM = ( ONE-ULP ) / SMLNUM
*
*     Compute 1-norm of each column of strictly upper triangular
*     part of T to control overflow in triangular solver.
*
      WORK( 1 ) = ZERO
      DO 30 J = 2, N
         WORK( J ) = ZERO
         DO 20 I = 1, J - 1
            WORK( J ) = WORK( J ) + ABS( T( I, J ) )
   20    CONTINUE
   30 CONTINUE
*
*     Index IP is used to specify the real or complex eigenvalue:
*       IP = 0, real eigenvalue,
*            1, first  of conjugate complex pair: (wr,wi)
*           -1, second of conjugate complex pair: (wr,wi)
*       ISCOMPLEX array stores IP for each column in current block.
*
      IF( RIGHTV ) THEN
*
*        ============================================================
*        Compute right eigenvectors.
*
*        IV is index of column in current block.
*        For complex right vector, uses IV-1 for real part and IV for complex part.
*        Non-blocked version always uses IV=2;
*        blocked     version starts with IV=NB, goes down to 1 or 2.
*        (Note the "0-th" column is used for 1-norms computed above.)
         IV = 2
         IF( NB.GT.2 ) THEN
            IV = NB
         END IF

         IP = 0
         IS = M
         DO 140 KI = N, 1, -1
            IF( IP.EQ.-1 ) THEN
*              previous iteration (ki+1) was second of conjugate pair,
*              so this ki is first of conjugate pair; skip to end of loop
               IP = 1
               GO TO 140
            ELSE IF( KI.EQ.1 ) THEN
*              last column, so this ki must be real eigenvalue
               IP = 0
            ELSE IF( T( KI, KI-1 ).EQ.ZERO ) THEN
*              zero on sub-diagonal, so this ki is real eigenvalue
               IP = 0
            ELSE
*              non-zero on sub-diagonal, so this ki is second of conjugate pair
               IP = -1
            END IF

            IF( SOMEV ) THEN
               IF( IP.EQ.0 ) THEN
                  IF( .NOT.SELECT( KI ) )
     $               GO TO 140
               ELSE
                  IF( .NOT.SELECT( KI-1 ) )
     $               GO TO 140
               END IF
            END IF
*
*           Compute the KI-th eigenvalue (WR,WI).
*
            WR = T( KI, KI )
            WI = ZERO
            IF( IP.NE.0 )
     $         WI = SQRT( ABS( T( KI, KI-1 ) ) )*
     $              SQRT( ABS( T( KI-1, KI ) ) )
            SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
            IF( IP.EQ.0 ) THEN
*
*              --------------------------------------------------------
*              Real right eigenvector
*
               WORK( KI + IV*N ) = ONE
*
*              Form right-hand side.
*
               DO 50 K = 1, KI - 1
                  WORK( K + IV*N ) = -T( K, KI )
   50          CONTINUE
*
*              Solve upper quasi-triangular system:
*              [ T(1:KI-1,1:KI-1) - WR ]*X = SCALE*WORK.
*
               JNXT = KI - 1
               DO 60 J = KI - 1, 1, -1
                  IF( J.GT.JNXT )
     $               GO TO 60
                  J1 = J
                  J2 = J
                  JNXT = J - 1
                  IF( J.GT.1 ) THEN
                     IF( T( J, J-1 ).NE.ZERO ) THEN
                        J1   = J - 1
                        JNXT = J - 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
                     CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale X(1,1) to avoid overflow when updating
*                    the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 )
                     WORK( J+IV*N ) = X( 1, 1 )
*
*                    Update right-hand side
*
                     CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
     $                           WORK( 1+IV*N ), 1 )
*
                  ELSE
*
*                    2-by-2 diagonal block
*
                     CALL DLALN2( .FALSE., 2, 1, SMIN, ONE,
     $                            T( J-1, J-1 ), LDT, ONE, ONE,
     $                            WORK( J-1+IV*N ), N, WR, ZERO, X, 2,
     $                            SCALE, XNORM, IERR )
*
*                    Scale X(1,1) and X(2,1) to avoid overflow when
*                    updating the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        BETA = MAX( WORK( J-1 ), WORK( J ) )
                        IF( BETA.GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           X( 2, 1 ) = X( 2, 1 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 )
                     WORK( J-1+IV*N ) = X( 1, 1 )
                     WORK( J  +IV*N ) = X( 2, 1 )
*
*                    Update right-hand side
*
                     CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
     $                           WORK( 1+IV*N ), 1 )
                     CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
     $                           WORK( 1+IV*N ), 1 )
                  END IF
   60          CONTINUE
*
*              Copy the vector x or Q*x to VR and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VR and normalize.
                  CALL DCOPY( KI, WORK( 1 + IV*N ), 1, VR( 1, IS ), 1 )
*
                  II = IDAMAX( KI, VR( 1, IS ), 1 )
                  REMAX = ONE / ABS( VR( II, IS ) )
                  CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
                  DO 70 K = KI + 1, N
                     VR( K, IS ) = ZERO
   70             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.GT.1 )
     $               CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
     $                           WORK( 1 + IV*N ), 1, WORK( KI + IV*N ),
     $                           VR( 1, KI ), 1 )
*
                  II = IDAMAX( N, VR( 1, KI ), 1 )
                  REMAX = ONE / ABS( VR( II, KI ) )
                  CALL DSCAL( N, REMAX, VR( 1, KI ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out below vector
                  DO K = KI + 1, N
                     WORK( K + IV*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV ) = IP
*                 back-transform and normalization is done below
               END IF
            ELSE
*
*              --------------------------------------------------------
*              Complex right eigenvector.
*
*              Initial solve
*              [ ( T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I*WI) ]*X = 0.
*              [ ( T(KI,  KI-1) T(KI,  KI) )               ]
*
               IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN
                  WORK( KI-1 + (IV-1)*N ) = ONE
                  WORK( KI   + (IV  )*N ) = WI / T( KI-1, KI )
               ELSE
                  WORK( KI-1 + (IV-1)*N ) = -WI / T( KI, KI-1 )
                  WORK( KI   + (IV  )*N ) = ONE
               END IF
               WORK( KI   + (IV-1)*N ) = ZERO
               WORK( KI-1 + (IV  )*N ) = ZERO
*
*              Form right-hand side.
*
               DO 80 K = 1, KI - 2
                  WORK( K+(IV-1)*N ) = -WORK( KI-1+(IV-1)*N )*T(K,KI-1)
                  WORK( K+(IV  )*N ) = -WORK( KI  +(IV  )*N )*T(K,KI  )
   80          CONTINUE
*
*              Solve upper quasi-triangular system:
*              [ T(1:KI-2,1:KI-2) - (WR+i*WI) ]*X = SCALE*(WORK+i*WORK2)
*
               JNXT = KI - 2
               DO 90 J = KI - 2, 1, -1
                  IF( J.GT.JNXT )
     $               GO TO 90
                  J1 = J
                  J2 = J
                  JNXT = J - 1
                  IF( J.GT.1 ) THEN
                     IF( T( J, J-1 ).NE.ZERO ) THEN
                        J1   = J - 1
                        JNXT = J - 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
                     CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+(IV-1)*N ), N,
     $                            WR, WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale X(1,1) and X(1,2) to avoid overflow when
*                    updating the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        IF( WORK( J ).GT.BIGNUM / XNORM ) THEN
                           X( 1, 1 ) = X( 1, 1 ) / XNORM
                           X( 1, 2 ) = X( 1, 2 ) / XNORM
                           SCALE = SCALE / XNORM
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ), 1 )
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV  )*N ), 1 )
                     END IF
                     WORK( J+(IV-1)*N ) = X( 1, 1 )
                     WORK( J+(IV  )*N ) = X( 1, 2 )
*
*                    Update the right-hand side
*
                     CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
     $                           WORK( 1+(IV-1)*N ), 1 )
                     CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
     $                           WORK( 1+(IV  )*N ), 1 )
*
                  ELSE
*
*                    2-by-2 diagonal block
*
                     CALL DLALN2( .FALSE., 2, 2, SMIN, ONE,
     $                            T( J-1, J-1 ), LDT, ONE, ONE,
     $                            WORK( J-1+(IV-1)*N ), N, WR, WI, X, 2,
     $                            SCALE, XNORM, IERR )
*
*                    Scale X to avoid overflow when updating
*                    the right-hand side.
*
                     IF( XNORM.GT.ONE ) THEN
                        BETA = MAX( WORK( J-1 ), WORK( J ) )
                        IF( BETA.GT.BIGNUM / XNORM ) THEN
                           REC = ONE / XNORM
                           X( 1, 1 ) = X( 1, 1 )*REC
                           X( 1, 2 ) = X( 1, 2 )*REC
                           X( 2, 1 ) = X( 2, 1 )*REC
                           X( 2, 2 ) = X( 2, 2 )*REC
                           SCALE = SCALE*REC
                        END IF
                     END IF
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ), 1 )
                        CALL DSCAL( KI, SCALE, WORK( 1+(IV  )*N ), 1 )
                     END IF
                     WORK( J-1+(IV-1)*N ) = X( 1, 1 )
                     WORK( J  +(IV-1)*N ) = X( 2, 1 )
                     WORK( J-1+(IV  )*N ) = X( 1, 2 )
                     WORK( J  +(IV  )*N ) = X( 2, 2 )
*
*                    Update the right-hand side
*
                     CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
     $                           WORK( 1+(IV-1)*N   ), 1 )
                     CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
     $                           WORK( 1+(IV-1)*N   ), 1 )
                     CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
     $                           WORK( 1+(IV  )*N ), 1 )
                     CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
     $                           WORK( 1+(IV  )*N ), 1 )
                  END IF
   90          CONTINUE
*
*              Copy the vector x or Q*x to VR and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VR and normalize.
                  CALL DCOPY( KI, WORK( 1+(IV-1)*N ), 1, VR(1,IS-1), 1 )
                  CALL DCOPY( KI, WORK( 1+(IV  )*N ), 1, VR(1,IS  ), 1 )
*
                  EMAX = ZERO
                  DO 100 K = 1, KI
                     EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+
     $                                 ABS( VR( K, IS   ) ) )
  100             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 )
                  CALL DSCAL( KI, REMAX, VR( 1, IS   ), 1 )
*
                  DO 110 K = KI + 1, N
                     VR( K, IS-1 ) = ZERO
                     VR( K, IS   ) = ZERO
  110             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.GT.2 ) THEN
                     CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
     $                           WORK( 1    + (IV-1)*N ), 1,
     $                           WORK( KI-1 + (IV-1)*N ), VR(1,KI-1), 1)
                     CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
     $                           WORK( 1  + (IV)*N ), 1,
     $                           WORK( KI + (IV)*N ), VR( 1, KI ), 1 )
                  ELSE
                     CALL DSCAL( N, WORK(KI-1+(IV-1)*N), VR(1,KI-1), 1)
                     CALL DSCAL( N, WORK(KI  +(IV  )*N), VR(1,KI  ), 1)
                  END IF
*
                  EMAX = ZERO
                  DO 120 K = 1, N
                     EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+
     $                                 ABS( VR( K, KI   ) ) )
  120             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 )
                  CALL DSCAL( N, REMAX, VR( 1, KI   ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out below vector
                  DO K = KI + 1, N
                     WORK( K + (IV-1)*N ) = ZERO
                     WORK( K + (IV  )*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV-1 ) = -IP
                  ISCOMPLEX( IV   ) =  IP
                  IV = IV - 1
*                 back-transform and normalization is done below
               END IF
            END IF

            IF( NB.GT.1 ) THEN
*              --------------------------------------------------------
*              Blocked version of back-transform
*              For complex case, KI2 includes both vectors (KI-1 and KI)
               IF( IP.EQ.0 ) THEN
                  KI2 = KI
               ELSE
                  KI2 = KI - 1
               END IF

*              Columns IV:NB of work are valid vectors.
*              When the number of vectors stored reaches NB-1 or NB,
*              or if this was last vector, do the GEMM
               IF( (IV.LE.2) .OR. (KI2.EQ.1) ) THEN
                  CALL DGEMM( 'N', 'N', N, NB-IV+1, KI2+NB-IV, ONE,
     $                        VR, LDVR,
     $                        WORK( 1 + (IV)*N    ), N,
     $                        ZERO,
     $                        WORK( 1 + (NB+IV)*N ), N )
*                 normalize vectors
                  DO K = IV, NB
                     IF( ISCOMPLEX(K).EQ.0 ) THEN
*                       real eigenvector
                        II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 )
                        REMAX = ONE / ABS( WORK( II + (NB+K)*N ) )
                     ELSE IF( ISCOMPLEX(K).EQ.1 ) THEN
*                       first eigenvector of conjugate pair
                        EMAX = ZERO
                        DO II = 1, N
                           EMAX = MAX( EMAX,
     $                                 ABS( WORK( II + (NB+K  )*N ) )+
     $                                 ABS( WORK( II + (NB+K+1)*N ) ) )
                        END DO
                        REMAX = ONE / EMAX
*                    else if ISCOMPLEX(K).EQ.-1
*                       second eigenvector of conjugate pair
*                       reuse same REMAX as previous K
                     END IF
                     CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 )
                  END DO
                  CALL DLACPY( 'F', N, NB-IV+1,
     $                         WORK( 1 + (NB+IV)*N ), N,
     $                         VR( 1, KI2 ), LDVR )
                  IV = NB
               ELSE
                  IV = IV - 1
               END IF
            END IF ! blocked back-transform
*
            IS = IS - 1
            IF( IP.NE.0 )
     $         IS = IS - 1
  140    CONTINUE
      END IF

      IF( LEFTV ) THEN
*
*        ============================================================
*        Compute left eigenvectors.
*
*        IV is index of column in current block.
*        For complex left vector, uses IV for real part and IV+1 for complex part.
*        Non-blocked version always uses IV=1;
*        blocked     version starts with IV=1, goes up to NB-1 or NB.
*        (Note the "0-th" column is used for 1-norms computed above.)
         IV = 1
         IP = 0
         IS = 1
         DO 260 KI = 1, N
            IF( IP.EQ.1 ) THEN
*              previous iteration (ki-1) was first of conjugate pair,
*              so this ki is second of conjugate pair; skip to end of loop
               IP = -1
               GO TO 260
            ELSE IF( KI.EQ.N ) THEN
*              last column, so this ki must be real eigenvalue
               IP = 0
            ELSE IF( T( KI+1, KI ).EQ.ZERO ) THEN
*              zero on sub-diagonal, so this ki is real eigenvalue
               IP = 0
            ELSE
*              non-zero on sub-diagonal, so this ki is first of conjugate pair
               IP = 1
            END IF
*
            IF( SOMEV ) THEN
               IF( .NOT.SELECT( KI ) )
     $            GO TO 260
            END IF
*
*           Compute the KI-th eigenvalue (WR,WI).
*
            WR = T( KI, KI )
            WI = ZERO
            IF( IP.NE.0 )
     $         WI = SQRT( ABS( T( KI, KI+1 ) ) )*
     $              SQRT( ABS( T( KI+1, KI ) ) )
            SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM )
*
            IF( IP.EQ.0 ) THEN
*
*              --------------------------------------------------------
*              Real left eigenvector
*
               WORK( KI + IV*N ) = ONE
*
*              Form right-hand side.
*
               DO 160 K = KI + 1, N
                  WORK( K + IV*N ) = -T( KI, K )
  160          CONTINUE
*
*              Solve transposed quasi-triangular system:
*              [ T(KI+1:N,KI+1:N) - WR ]**T * X = SCALE*WORK
*
               VMAX = ONE
               VCRIT = BIGNUM
*
               JNXT = KI + 1
               DO 170 J = KI + 1, N
                  IF( J.LT.JNXT )
     $               GO TO 170
                  J1 = J
                  J2 = J
                  JNXT = J + 1
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).NE.ZERO ) THEN
                        J2 = J + 1
                        JNXT = J + 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side.
*
                     IF( WORK( J ).GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+IV*N ) = WORK( J+IV*N ) -
     $                                DDOT( J-KI-1, T( KI+1, J ), 1,
     $                                      WORK( KI+1+IV*N ), 1 )
*
*                    Solve [ T(J,J) - WR ]**T * X = WORK
*
                     CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ), 1 )
                     WORK( J+IV*N ) = X( 1, 1 )
                     VMAX = MAX( ABS( WORK( J+IV*N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  ELSE
*
*                    2-by-2 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side.
*
                     BETA = MAX( WORK( J ), WORK( J+1 ) )
                     IF( BETA.GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+IV*N ) = WORK( J+IV*N ) -
     $                                DDOT( J-KI-1, T( KI+1, J ), 1,
     $                                      WORK( KI+1+IV*N ), 1 )
*
                     WORK( J+1+IV*N ) = WORK( J+1+IV*N ) -
     $                                  DDOT( J-KI-1, T( KI+1, J+1 ), 1,
     $                                        WORK( KI+1+IV*N ), 1 )
*
*                    Solve
*                    [ T(J,J)-WR   T(J,J+1)      ]**T * X = SCALE*( WORK1 )
*                    [ T(J+1,J)    T(J+1,J+1)-WR ]                ( WORK2 )
*
                     CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            ZERO, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE )
     $                  CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ), 1 )
                     WORK( J  +IV*N ) = X( 1, 1 )
                     WORK( J+1+IV*N ) = X( 2, 1 )
*
                     VMAX = MAX( ABS( WORK( J  +IV*N ) ),
     $                           ABS( WORK( J+1+IV*N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  END IF
  170          CONTINUE
*
*              Copy the vector x or Q*x to VL and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VL and normalize.
                  CALL DCOPY( N-KI+1, WORK( KI + IV*N ), 1,
     $                                VL( KI, IS ), 1 )
*
                  II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
                  REMAX = ONE / ABS( VL( II, IS ) )
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
                  DO 180 K = 1, KI - 1
                     VL( K, IS ) = ZERO
  180             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.LT.N )
     $               CALL DGEMV( 'N', N, N-KI, ONE,
     $                           VL( 1, KI+1 ), LDVL,
     $                           WORK( KI+1 + IV*N ), 1,
     $                           WORK( KI   + IV*N ), VL( 1, KI ), 1 )
*
                  II = IDAMAX( N, VL( 1, KI ), 1 )
                  REMAX = ONE / ABS( VL( II, KI ) )
                  CALL DSCAL( N, REMAX, VL( 1, KI ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out above vector
*                 could go from KI-NV+1 to KI-1
                  DO K = 1, KI - 1
                     WORK( K + IV*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV ) = IP
*                 back-transform and normalization is done below
               END IF
            ELSE
*
*              --------------------------------------------------------
*              Complex left eigenvector.
*
*              Initial solve:
*              [ ( T(KI,KI)    T(KI,KI+1)  )**T - (WR - I* WI) ]*X = 0.
*              [ ( T(KI+1,KI) T(KI+1,KI+1) )                   ]
*
               IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN
                  WORK( KI   + (IV  )*N ) = WI / T( KI, KI+1 )
                  WORK( KI+1 + (IV+1)*N ) = ONE
               ELSE
                  WORK( KI   + (IV  )*N ) = ONE
                  WORK( KI+1 + (IV+1)*N ) = -WI / T( KI+1, KI )
               END IF
               WORK( KI+1 + (IV  )*N ) = ZERO
               WORK( KI   + (IV+1)*N ) = ZERO
*
*              Form right-hand side.
*
               DO 190 K = KI + 2, N
                  WORK( K+(IV  )*N ) = -WORK( KI  +(IV  )*N )*T(KI,  K)
                  WORK( K+(IV+1)*N ) = -WORK( KI+1+(IV+1)*N )*T(KI+1,K)
  190          CONTINUE
*
*              Solve transposed quasi-triangular system:
*              [ T(KI+2:N,KI+2:N)**T - (WR-i*WI) ]*X = WORK1+i*WORK2
*
               VMAX = ONE
               VCRIT = BIGNUM
*
               JNXT = KI + 2
               DO 200 J = KI + 2, N
                  IF( J.LT.JNXT )
     $               GO TO 200
                  J1 = J
                  J2 = J
                  JNXT = J + 1
                  IF( J.LT.N ) THEN
                     IF( T( J+1, J ).NE.ZERO ) THEN
                        J2 = J + 1
                        JNXT = J + 2
                     END IF
                  END IF
*
                  IF( J1.EQ.J2 ) THEN
*
*                    1-by-1 diagonal block
*
*                    Scale if necessary to avoid overflow when
*                    forming the right-hand side elements.
*
                     IF( WORK( J ).GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV  )*N), 1 )
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J+(IV  )*N ) = WORK( J+(IV)*N ) -
     $                                  DDOT( J-KI-2, T( KI+2, J ), 1,
     $                                        WORK( KI+2+(IV)*N ), 1 )
                     WORK( J+(IV+1)*N ) = WORK( J+(IV+1)*N ) -
     $                                  DDOT( J-KI-2, T( KI+2, J ), 1,
     $                                        WORK( KI+2+(IV+1)*N ), 1 )
*
*                    Solve [ T(J,J)-(WR-i*WI) ]*(X11+i*X12)= WK+I*WK2
*
                     CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            -WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV  )*N), 1)
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N), 1)
                     END IF
                     WORK( J+(IV  )*N ) = X( 1, 1 )
                     WORK( J+(IV+1)*N ) = X( 1, 2 )
                     VMAX = MAX( ABS( WORK( J+(IV  )*N ) ),
     $                           ABS( WORK( J+(IV+1)*N ) ), VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  ELSE
*
*                    2-by-2 diagonal block
*
*                    Scale if necessary to avoid overflow when forming
*                    the right-hand side elements.
*
                     BETA = MAX( WORK( J ), WORK( J+1 ) )
                     IF( BETA.GT.VCRIT ) THEN
                        REC = ONE / VMAX
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV  )*N), 1 )
                        CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N), 1 )
                        VMAX = ONE
                        VCRIT = BIGNUM
                     END IF
*
                     WORK( J  +(IV  )*N ) = WORK( J+(IV)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J ), 1,
     $                                      WORK( KI+2+(IV)*N ), 1 )
*
                     WORK( J  +(IV+1)*N ) = WORK( J+(IV+1)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J ), 1,
     $                                      WORK( KI+2+(IV+1)*N ), 1 )
*
                     WORK( J+1+(IV  )*N ) = WORK( J+1+(IV)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J+1 ), 1,
     $                                      WORK( KI+2+(IV)*N ), 1 )
*
                     WORK( J+1+(IV+1)*N ) = WORK( J+1+(IV+1)*N ) -
     $                                DDOT( J-KI-2, T( KI+2, J+1 ), 1,
     $                                      WORK( KI+2+(IV+1)*N ), 1 )
*
*                    Solve 2-by-2 complex linear equation
*                    [ (T(j,j)   T(j,j+1)  )**T - (wr-i*wi)*I ]*X = SCALE*B
*                    [ (T(j+1,j) T(j+1,j+1))                  ]
*
                     CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ),
     $                            LDT, ONE, ONE, WORK( J+IV*N ), N, WR,
     $                            -WI, X, 2, SCALE, XNORM, IERR )
*
*                    Scale if necessary
*
                     IF( SCALE.NE.ONE ) THEN
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV  )*N), 1)
                        CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N), 1)
                     END IF
                     WORK( J  +(IV  )*N ) = X( 1, 1 )
                     WORK( J  +(IV+1)*N ) = X( 1, 2 )
                     WORK( J+1+(IV  )*N ) = X( 2, 1 )
                     WORK( J+1+(IV+1)*N ) = X( 2, 2 )
                     VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ),
     $                           ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ),
     $                           VMAX )
                     VCRIT = BIGNUM / VMAX
*
                  END IF
  200          CONTINUE
*
*              Copy the vector x or Q*x to VL and normalize.
*
               IF( .NOT.OVER ) THEN
*                 ------------------------------
*                 no back-transform: copy x to VL and normalize.
                  CALL DCOPY( N-KI+1, WORK( KI + (IV  )*N ), 1,
     $                        VL( KI, IS   ), 1 )
                  CALL DCOPY( N-KI+1, WORK( KI + (IV+1)*N ), 1,
     $                        VL( KI, IS+1 ), 1 )
*
                  EMAX = ZERO
                  DO 220 K = KI, N
                     EMAX = MAX( EMAX, ABS( VL( K, IS   ) )+
     $                                 ABS( VL( K, IS+1 ) ) )
  220             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS   ), 1 )
                  CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 )
*
                  DO 230 K = 1, KI - 1
                     VL( K, IS   ) = ZERO
                     VL( K, IS+1 ) = ZERO
  230             CONTINUE
*
               ELSE IF( NB.EQ.1 ) THEN
*                 ------------------------------
*                 version 1: back-transform each vector with GEMV, Q*x.
                  IF( KI.LT.N-1 ) THEN
                     CALL DGEMV( 'N', N, N-KI-1, ONE,
     $                           VL( 1, KI+2 ), LDVL,
     $                           WORK( KI+2 + (IV)*N ), 1,
     $                           WORK( KI   + (IV)*N ),
     $                           VL( 1, KI ), 1 )
                     CALL DGEMV( 'N', N, N-KI-1, ONE,
     $                           VL( 1, KI+2 ), LDVL,
     $                           WORK( KI+2 + (IV+1)*N ), 1,
     $                           WORK( KI+1 + (IV+1)*N ),
     $                           VL( 1, KI+1 ), 1 )
                  ELSE
                     CALL DSCAL( N, WORK(KI+  (IV  )*N), VL(1, KI  ), 1)
                     CALL DSCAL( N, WORK(KI+1+(IV+1)*N), VL(1, KI+1), 1)
                  END IF
*
                  EMAX = ZERO
                  DO 240 K = 1, N
                     EMAX = MAX( EMAX, ABS( VL( K, KI   ) )+
     $                                 ABS( VL( K, KI+1 ) ) )
  240             CONTINUE
                  REMAX = ONE / EMAX
                  CALL DSCAL( N, REMAX, VL( 1, KI   ), 1 )
                  CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 )
*
               ELSE
*                 ------------------------------
*                 version 2: back-transform block of vectors with GEMM
*                 zero out above vector
*                 could go from KI-NV+1 to KI-1
                  DO K = 1, KI - 1
                     WORK( K + (IV  )*N ) = ZERO
                     WORK( K + (IV+1)*N ) = ZERO
                  END DO
                  ISCOMPLEX( IV   ) =  IP
                  ISCOMPLEX( IV+1 ) = -IP
                  IV = IV + 1
*                 back-transform and normalization is done below
               END IF
            END IF

            IF( NB.GT.1 ) THEN
*              --------------------------------------------------------
*              Blocked version of back-transform
*              For complex case, KI2 includes both vectors (KI and KI+1)
               IF( IP.EQ.0 ) THEN
                  KI2 = KI
               ELSE
                  KI2 = KI + 1
               END IF

*              Columns 1:IV of work are valid vectors.
*              When the number of vectors stored reaches NB-1 or NB,
*              or if this was last vector, do the GEMM
               IF( (IV.GE.NB-1) .OR. (KI2.EQ.N) ) THEN
                  CALL DGEMM( 'N', 'N', N, IV, N-KI2+IV, ONE,
     $                        VL( 1, KI2-IV+1 ), LDVL,
     $                        WORK( KI2-IV+1 + (1)*N ), N,
     $                        ZERO,
     $                        WORK( 1 + (NB+1)*N ), N )
*                 normalize vectors
                  DO K = 1, IV
                     IF( ISCOMPLEX(K).EQ.0) THEN
*                       real eigenvector
                        II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 )
                        REMAX = ONE / ABS( WORK( II + (NB+K)*N ) )
                     ELSE IF( ISCOMPLEX(K).EQ.1) THEN
*                       first eigenvector of conjugate pair
                        EMAX = ZERO
                        DO II = 1, N
                           EMAX = MAX( EMAX,
     $                                 ABS( WORK( II + (NB+K  )*N ) )+
     $                                 ABS( WORK( II + (NB+K+1)*N ) ) )
                        END DO
                        REMAX = ONE / EMAX
*                    else if ISCOMPLEX(K).EQ.-1
*                       second eigenvector of conjugate pair
*                       reuse same REMAX as previous K
                     END IF
                     CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 )
                  END DO
                  CALL DLACPY( 'F', N, IV,
     $                         WORK( 1 + (NB+1)*N ), N,
     $                         VL( 1, KI2-IV+1 ), LDVL )
                  IV = 1
               ELSE
                  IV = IV + 1
               END IF
            END IF ! blocked back-transform
*
            IS = IS + 1
            IF( IP.NE.0 )
     $         IS = IS + 1
  260    CONTINUE
      END IF
*
      RETURN
*
*     End of DTREVC3
*
      END