cgeevx.f

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*> \brief <b> CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*> \htmlonly
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*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
*                          LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
*                          RCONDV, WORK, LWORK, RWORK, INFO )
* 
*       .. Scalar Arguments ..
*       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
*       INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
*       REAL               ABNRM
*       ..
*       .. Array Arguments ..
*       REAL               RCONDE( * ), RCONDV( * ), RWORK( * ),
*      $                   SCALE( * )
*       COMPLEX            A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   W( * ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
*> eigenvalues and, optionally, the left and/or right eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
*> (RCONDE), and reciprocal condition numbers for the right
*> eigenvectors (RCONDV).
*>
*> The right eigenvector v(j) of A satisfies
*>                  A * v(j) = lambda(j) * v(j)
*> where lambda(j) is its eigenvalue.
*> The left eigenvector u(j) of A satisfies
*>               u(j)**H * A = lambda(j) * u(j)**H
*> where u(j)**H denotes the conjugate transpose of u(j).
*>
*> The computed eigenvectors are normalized to have Euclidean norm
*> equal to 1 and largest component real.
*>
*> Balancing a matrix means permuting the rows and columns to make it
*> more nearly upper triangular, and applying a diagonal similarity
*> transformation D * A * D**(-1), where D is a diagonal matrix, to
*> make its rows and columns closer in norm and the condition numbers
*> of its eigenvalues and eigenvectors smaller.  The computed
*> reciprocal condition numbers correspond to the balanced matrix.
*> Permuting rows and columns will not change the condition numbers
*> (in exact arithmetic) but diagonal scaling will.  For further
*> explanation of balancing, see section 4.10.2 of the LAPACK
*> Users' Guide.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] BALANC
*> \verbatim
*>          BALANC is CHARACTER*1
*>          Indicates how the input matrix should be diagonally scaled
*>          and/or permuted to improve the conditioning of its
*>          eigenvalues.
*>          = 'N': Do not diagonally scale or permute;
*>          = 'P': Perform permutations to make the matrix more nearly
*>                 upper triangular. Do not diagonally scale;
*>          = 'S': Diagonally scale the matrix, ie. replace A by
*>                 D*A*D**(-1), where D is a diagonal matrix chosen
*>                 to make the rows and columns of A more equal in
*>                 norm. Do not permute;
*>          = 'B': Both diagonally scale and permute A.
*>
*>          Computed reciprocal condition numbers will be for the matrix
*>          after balancing and/or permuting. Permuting does not change
*>          condition numbers (in exact arithmetic), but balancing does.
*> \endverbatim
*>
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N': left eigenvectors of A are not computed;
*>          = 'V': left eigenvectors of A are computed.
*>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N': right eigenvectors of A are not computed;
*>          = 'V': right eigenvectors of A are computed.
*>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          SENSE is CHARACTER*1
*>          Determines which reciprocal condition numbers are computed.
*>          = 'N': None are computed;
*>          = 'E': Computed for eigenvalues only;
*>          = 'V': Computed for right eigenvectors only;
*>          = 'B': Computed for eigenvalues and right eigenvectors.
*>
*>          If SENSE = 'E' or 'B', both left and right eigenvectors
*>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX array, dimension (LDA,N)
*>          On entry, the N-by-N matrix A.
*>          On exit, A has been overwritten.  If JOBVL = 'V' or
*>          JOBVR = 'V', A contains the Schur form of the balanced 
*>          version of the matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is COMPLEX array, dimension (N)
*>          W contains the computed eigenvalues.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is COMPLEX array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order
*>          as their eigenvalues.
*>          If JOBVL = 'N', VL is not referenced.
*>          u(j) = VL(:,j), the j-th column of VL.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.  LDVL >= 1; if
*>          JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is COMPLEX array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order
*>          as their eigenvalues.
*>          If JOBVR = 'N', VR is not referenced.
*>          v(j) = VR(:,j), the j-th column of VR.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1; if
*>          JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI are integer values determined when A was
*>          balanced.  The balanced A(i,j) = 0 if I > J and
*>          J = 1,...,ILO-1 or I = IHI+1,...,N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is REAL array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          when balancing A.  If P(j) is the index of the row and column
*>          interchanged with row and column j, and D(j) is the scaling
*>          factor applied to row and column j, then
*>          SCALE(J) = P(J),    for J = 1,...,ILO-1
*>                   = D(J),    for J = ILO,...,IHI
*>                   = P(J)     for J = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*>          ABNRM is REAL
*>          The one-norm of the balanced matrix (the maximum
*>          of the sum of absolute values of elements of any column).
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is REAL array, dimension (N)
*>          RCONDE(j) is the reciprocal condition number of the j-th
*>          eigenvalue.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is REAL array, dimension (N)
*>          RCONDV(j) is the reciprocal condition number of the j-th
*>          right eigenvector.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  If SENSE = 'N' or 'E',
*>          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
*>          LWORK >= N*N+2*N.
*>          For good performance, LWORK must generally be larger.
*>
*>          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] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = i, the QR algorithm failed to compute all the
*>                eigenvalues, and no eigenvectors or condition numbers
*>                have been computed; elements 1:ILO-1 and i+1:N of W
*>                contain eigenvalues which have converged.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup complexGEeigen
*
*  =====================================================================
      SUBROUTINE cgeevx( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
     $                   ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde,
     $                   rcondv, work, lwork, rwork, info )
*
*  -- LAPACK driver routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      CHARACTER          balanc, jobvl, jobvr, sense
      INTEGER            ihi, ilo, info, lda, ldvl, ldvr, lwork, n
      REAL               abnrm
*     ..
*     .. Array Arguments ..
      REAL               rconde( * ), rcondv( * ), rwork( * ),
     $                   scale( * )
      COMPLEX            a( lda, * ), vl( ldvl, * ), vr( ldvr, * ),
     $                   w( * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               zero, one
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            lquery, scalea, wantvl, wantvr, wntsnb, wntsne,
     $                   wntsnn, wntsnv
      CHARACTER          job, side
      INTEGER            hswork, i, icond, ierr, itau, iwrk, k, maxwrk,
     $                   minwrk, nout
      REAL               anrm, bignum, cscale, eps, scl, smlnum
      COMPLEX            tmp
*     ..
*     .. Local Arrays ..
      LOGICAL            select( 1 )
      REAL               dum( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           cgebak, cgebal, cgehrd, chseqr, clacpy, clascl,
     $                   cscal, csscal, ctrevc, ctrsna, cunghr, slabad,
     $                   slascl, xerbla
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      INTEGER            ilaenv, isamax
      REAL               clange, scnrm2, slamch
      EXTERNAL           lsame, ilaenv, isamax, clange, scnrm2, slamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          aimag, cmplx, conjg, max, REAL, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      lquery = ( lwork.EQ.-1 )
      wantvl = lsame( jobvl, 'V' )
      wantvr = lsame( jobvr, 'V' )
      wntsnn = lsame( sense, 'N' )
      wntsne = lsame( sense, 'E' )
      wntsnv = lsame( sense, 'V' )
      wntsnb = lsame( sense, 'B' )
      IF( .NOT.( lsame( balanc, 'N' ) .OR. lsame( balanc, 'S' ) .OR.
     $    lsame( balanc, 'P' ) .OR. lsame( balanc, 'B' ) ) ) THEN
         info = -1
      ELSE IF( ( .NOT.wantvl ) .AND. ( .NOT.lsame( jobvl, 'N' ) ) ) THEN
         info = -2
      ELSE IF( ( .NOT.wantvr ) .AND. ( .NOT.lsame( jobvr, 'N' ) ) ) THEN
         info = -3
      ELSE IF( .NOT.( wntsnn .OR. wntsne .OR. wntsnb .OR. wntsnv ) .OR.
     $         ( ( wntsne .OR. wntsnb ) .AND. .NOT.( wantvl .AND.
     $         wantvr ) ) ) THEN
         info = -4
      ELSE IF( n.LT.0 ) THEN
         info = -5
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -7
      ELSE IF( ldvl.LT.1 .OR. ( wantvl .AND. ldvl.LT.n ) ) THEN
         info = -10
      ELSE IF( ldvr.LT.1 .OR. ( wantvr .AND. ldvr.LT.n ) ) THEN
         info = -12
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       CWorkspace refers to complex workspace, and RWorkspace to real
*       workspace. NB refers to the optimal block size for the
*       immediately following subroutine, as returned by ILAENV.
*       HSWORK refers to the workspace preferred by CHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.)
*
      IF( info.EQ.0 ) THEN
         IF( n.EQ.0 ) THEN
            minwrk = 1
            maxwrk = 1
         ELSE
            maxwrk = n + n*ilaenv( 1, 'CGEHRD', ' ', n, 1, n, 0 )
*
            IF( wantvl ) THEN
               CALL chseqr( 'S', 'V', n, 1, n, a, lda, w, vl, ldvl,
     $                work, -1, info )
            ELSE IF( wantvr ) THEN
               CALL chseqr( 'S', 'V', n, 1, n, a, lda, w, vr, ldvr,
     $                work, -1, info )
            ELSE
               IF( wntsnn ) THEN
                  CALL chseqr( 'E', 'N', n, 1, n, a, lda, w, vr, ldvr,
     $                work, -1, info )
               ELSE
                  CALL chseqr( 'S', 'N', n, 1, n, a, lda, w, vr, ldvr,
     $                work, -1, info )
               END IF
            END IF
            hswork = work( 1 )
*
            IF( ( .NOT.wantvl ) .AND. ( .NOT.wantvr ) ) THEN
               minwrk = 2*n
               IF( .NOT.( wntsnn .OR. wntsne ) )
     $            minwrk = max( minwrk, n*n + 2*n )
               maxwrk = max( maxwrk, hswork )
               IF( .NOT.( wntsnn .OR. wntsne ) )
     $            maxwrk = max( maxwrk, n*n + 2*n )
            ELSE
               minwrk = 2*n
               IF( .NOT.( wntsnn .OR. wntsne ) )
     $            minwrk = max( minwrk, n*n + 2*n )
               maxwrk = max( maxwrk, hswork )
               maxwrk = max( maxwrk, n + ( n - 1 )*ilaenv( 1, 'CUNGHR',
     $                       ' ', n, 1, n, -1 ) )
               IF( .NOT.( wntsnn .OR. wntsne ) )
     $            maxwrk = max( maxwrk, n*n + 2*n )
               maxwrk = max( maxwrk, 2*n )
            END IF
            maxwrk = max( maxwrk, minwrk )
         END IF
         work( 1 ) = maxwrk
*
         IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
            info = -20
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'CGEEVX', -info )
         return
      ELSE IF( lquery ) THEN
         return
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   return
*
*     Get machine constants
*
      eps = slamch( 'P' )
      smlnum = slamch( 'S' )
      bignum = one / smlnum
      CALL slabad( smlnum, bignum )
      smlnum = sqrt( smlnum ) / eps
      bignum = one / smlnum
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      icond = 0
      anrm = clange( 'M', n, n, a, lda, dum )
      scalea = .false.
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
         scalea = .true.
         cscale = smlnum
      ELSE IF( anrm.GT.bignum ) THEN
         scalea = .true.
         cscale = bignum
      END IF
      IF( scalea )
     $   CALL clascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
*
*     Balance the matrix and compute ABNRM
*
      CALL cgebal( balanc, n, a, lda, ilo, ihi, scale, ierr )
      abnrm = clange( '1', n, n, a, lda, dum )
      IF( scalea ) THEN
         dum( 1 ) = abnrm
         CALL slascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
         abnrm = dum( 1 )
      END IF
*
*     Reduce to upper Hessenberg form
*     (CWorkspace: need 2*N, prefer N+N*NB)
*     (RWorkspace: none)
*
      itau = 1
      iwrk = itau + n
      CALL cgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
     $             lwork-iwrk+1, ierr )
*
      IF( wantvl ) THEN
*
*        Want left eigenvectors
*        Copy Householder vectors to VL
*
         side = 'L'
         CALL clacpy( 'L', n, n, a, lda, vl, ldvl )
*
*        Generate unitary matrix in VL
*        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
*        (RWorkspace: none)
*
         CALL cunghr( n, ilo, ihi, vl, ldvl, work( itau ), work( iwrk ),
     $                lwork-iwrk+1, ierr )
*
*        Perform QR iteration, accumulating Schur vectors in VL
*        (CWorkspace: need 1, prefer HSWORK (see comments) )
*        (RWorkspace: none)
*
         iwrk = itau
         CALL chseqr( 'S', 'V', n, ilo, ihi, a, lda, w, vl, ldvl,
     $                work( iwrk ), lwork-iwrk+1, info )
*
         IF( wantvr ) THEN
*
*           Want left and right eigenvectors
*           Copy Schur vectors to VR
*
            side = 'B'
            CALL clacpy( 'F', n, n, vl, ldvl, vr, ldvr )
         END IF
*
      ELSE IF( wantvr ) THEN
*
*        Want right eigenvectors
*        Copy Householder vectors to VR
*
         side = 'R'
         CALL clacpy( 'L', n, n, a, lda, vr, ldvr )
*
*        Generate unitary matrix in VR
*        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
*        (RWorkspace: none)
*
         CALL cunghr( n, ilo, ihi, vr, ldvr, work( itau ), work( iwrk ),
     $                lwork-iwrk+1, ierr )
*
*        Perform QR iteration, accumulating Schur vectors in VR
*        (CWorkspace: need 1, prefer HSWORK (see comments) )
*        (RWorkspace: none)
*
         iwrk = itau
         CALL chseqr( 'S', 'V', n, ilo, ihi, a, lda, w, vr, ldvr,
     $                work( iwrk ), lwork-iwrk+1, info )
*
      ELSE
*
*        Compute eigenvalues only
*        If condition numbers desired, compute Schur form
*
         IF( wntsnn ) THEN
            job = 'E'
         ELSE
            job = 'S'
         END IF
*
*        (CWorkspace: need 1, prefer HSWORK (see comments) )
*        (RWorkspace: none)
*
         iwrk = itau
         CALL chseqr( job, 'N', n, ilo, ihi, a, lda, w, vr, ldvr,
     $                work( iwrk ), lwork-iwrk+1, info )
      END IF
*
*     If INFO > 0 from CHSEQR, then quit
*
      IF( info.GT.0 )
     $   go to 50
*
      IF( wantvl .OR. wantvr ) THEN
*
*        Compute left and/or right eigenvectors
*        (CWorkspace: need 2*N)
*        (RWorkspace: need N)
*
         CALL ctrevc( side, 'B', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
     $                n, nout, work( iwrk ), rwork, ierr )
      END IF
*
*     Compute condition numbers if desired
*     (CWorkspace: need N*N+2*N unless SENSE = 'E')
*     (RWorkspace: need 2*N unless SENSE = 'E')
*
      IF( .NOT.wntsnn ) THEN
         CALL ctrsna( sense, 'A', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
     $                rconde, rcondv, n, nout, work( iwrk ), n, rwork,
     $                icond )
      END IF
*
      IF( wantvl ) THEN
*
*        Undo balancing of left eigenvectors
*
         CALL cgebak( balanc, 'L', n, ilo, ihi, scale, n, vl, ldvl,
     $                ierr )
*
*        Normalize left eigenvectors and make largest component real
*
         DO 20 i = 1, n
            scl = one / scnrm2( n, vl( 1, i ), 1 )
            CALL csscal( n, scl, vl( 1, i ), 1 )
            DO 10 k = 1, n
               rwork( k ) = REAL( VL( K, I ) )**2 +
     $                      aimag( vl( k, i ) )**2
   10       continue
            k = isamax( n, rwork, 1 )
            tmp = conjg( vl( k, i ) ) / sqrt( rwork( k ) )
            CALL cscal( n, tmp, vl( 1, i ), 1 )
            vl( k, i ) = cmplx( REAL( VL( K, I ) ), zero )
   20    continue
      END IF
*
      IF( wantvr ) THEN
*
*        Undo balancing of right eigenvectors
*
         CALL cgebak( balanc, 'R', n, ilo, ihi, scale, n, vr, ldvr,
     $                ierr )
*
*        Normalize right eigenvectors and make largest component real
*
         DO 40 i = 1, n
            scl = one / scnrm2( n, vr( 1, i ), 1 )
            CALL csscal( n, scl, vr( 1, i ), 1 )
            DO 30 k = 1, n
               rwork( k ) = REAL( VR( K, I ) )**2 +
     $                      aimag( vr( k, i ) )**2
   30       continue
            k = isamax( n, rwork, 1 )
            tmp = conjg( vr( k, i ) ) / sqrt( rwork( k ) )
            CALL cscal( n, tmp, vr( 1, i ), 1 )
            vr( k, i ) = cmplx( REAL( VR( K, I ) ), zero )
   40    continue
      END IF
*
*     Undo scaling if necessary
*
   50 continue
      IF( scalea ) THEN
         CALL clascl( 'G', 0, 0, cscale, anrm, n-info, 1, w( info+1 ),
     $                max( n-info, 1 ), ierr )
         IF( info.EQ.0 ) THEN
            IF( ( wntsnv .OR. wntsnb ) .AND. icond.EQ.0 )
     $         CALL slascl( 'G', 0, 0, cscale, anrm, n, 1, rcondv, n,
     $                      ierr )
         ELSE
            CALL clascl( 'G', 0, 0, cscale, anrm, ilo-1, 1, w, n, ierr )
         END IF
      END IF
*
      work( 1 ) = maxwrk
      return
*
*     End of CGEEVX
*
      END