zgees.f

          SUBROUTINE ZGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS,
         $                  LDVS, WORK, LWORK, RWORK, BWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBVS, SORT
          INTEGER            INFO, LDA, LDVS, LWORK, N, SDIM
    *     ..
    *     .. Array Arguments ..
          LOGICAL            BWORK( * )
          DOUBLE PRECISION   RWORK( * )
          COMPLEX*16         A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
    *     ..
    *     .. Function Arguments ..
          LOGICAL            SELECT
          EXTERNAL           SELECT
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZGEES computes for an N-by-N complex nonsymmetric matrix A, the
    *  eigenvalues, the Schur form T, and, optionally, the matrix of Schur
    *  vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
    *
    *  Optionally, it also orders the eigenvalues on the diagonal of the
    *  Schur form so that selected eigenvalues are at the top left.
    *  The leading columns of Z then form an orthonormal basis for the
    *  invariant subspace corresponding to the selected eigenvalues.
    *
    *  A complex matrix is in Schur form if it is upper triangular.
    *
    *  Arguments
    *  =========
    *
    *  JOBVS   (input) CHARACTER*1
    *          = 'N': Schur vectors are not computed;
    *          = 'V': Schur vectors are computed.
    *
    *  SORT    (input) CHARACTER*1
    *          Specifies whether or not to order the eigenvalues on the
    *          diagonal of the Schur form.
    *          = 'N': Eigenvalues are not ordered:
    *          = 'S': Eigenvalues are ordered (see SELECT).
    *
    *  SELECT  (external procedure) LOGICAL FUNCTION of one COMPLEX*16 argument
    *          SELECT must be declared EXTERNAL in the calling subroutine.
    *          If SORT = 'S', SELECT is used to select eigenvalues to order
    *          to the top left of the Schur form.
    *          IF SORT = 'N', SELECT is not referenced.
    *          The eigenvalue W(j) is selected if SELECT(W(j)) is true.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A. N >= 0.
    *
    *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
    *          On entry, the N-by-N matrix A.
    *          On exit, A has been overwritten by its Schur form T.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  SDIM    (output) INTEGER
    *          If SORT = 'N', SDIM = 0.
    *          If SORT = 'S', SDIM = number of eigenvalues for which
    *                         SELECT is true.
    *
    *  W       (output) COMPLEX*16 array, dimension (N)
    *          W contains the computed eigenvalues, in the same order that
    *          they appear on the diagonal of the output Schur form T.
    *
    *  VS      (output) COMPLEX*16 array, dimension (LDVS,N)
    *          If JOBVS = 'V', VS contains the unitary matrix Z of Schur
    *          vectors.
    *          If JOBVS = 'N', VS is not referenced.
    *
    *  LDVS    (input) INTEGER
    *          The leading dimension of the array VS.  LDVS >= 1; if
    *          JOBVS = 'V', LDVS >= N.
    *
    *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK.  LWORK >= max(1,2*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.
    *
    *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
    *
    *  BWORK   (workspace) LOGICAL array, dimension (N)
    *          Not referenced if SORT = 'N'.
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value.
    *          > 0: if INFO = i, and i is
    *               <= N:  the QR algorithm failed to compute all the
    *                      eigenvalues; elements 1:ILO-1 and i+1:N of W
    *                      contain those eigenvalues which have converged;
    *                      if JOBVS = 'V', VS contains the matrix which
    *                      reduces A to its partially converged Schur form.
    *               = N+1: the eigenvalues could not be reordered because
    *                      some eigenvalues were too close to separate (the
    *                      problem is very ill-conditioned);
    *               = N+2: after reordering, roundoff changed values of
    *                      some complex eigenvalues so that leading
    *                      eigenvalues in the Schur form no longer satisfy
    *                      SELECT = .TRUE..  This could also be caused by
    *                      underflow due to scaling.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY, SCALEA, WANTST, WANTVS
          INTEGER            HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
         $                   ITAU, IWRK, MAXWRK, MINWRK
          DOUBLE PRECISION   ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM
    *     ..
    *     .. Local Arrays ..
          DOUBLE PRECISION   DUM( 1 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLABAD, XERBLA, ZCOPY, ZGEBAK, ZGEBAL, ZGEHRD,
         $                   ZHSEQR, ZLACPY, ZLASCL, ZTRSEN, ZUNGHR
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          DOUBLE PRECISION   DLAMCH, ZLANGE
          EXTERNAL           LSAME, ILAENV, DLAMCH, ZLANGE
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input arguments
    *
          INFO = 0
          LQUERY = ( LWORK.EQ.-1 )
          WANTVS = LSAME( JOBVS, 'V' )
          WANTST = LSAME( SORT, 'S' )
          IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
             INFO = -1
          ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -4
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -6
          ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
             INFO = -10
          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 ZHSEQR, 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, 'ZGEHRD', ' ', N, 1, N, 0 )
                MINWRK = 2*N
    *
                CALL ZHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
         $             WORK, -1, IEVAL )
                HSWORK = WORK( 1 )
    *
                IF( .NOT.WANTVS ) THEN
                   MAXWRK = MAX( MAXWRK, HSWORK )
                ELSE
                   MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
         $                       ' ', N, 1, N, -1 ) )
                   MAXWRK = MAX( MAXWRK, HSWORK )
                END IF
             END IF
             WORK( 1 ) = MAXWRK
    *
             IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
                INFO = -12
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZGEES ', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 ) THEN
             SDIM = 0
             RETURN
          END IF
    *
    *     Get machine constants
    *
          EPS = DLAMCH( 'P' )
          SMLNUM = DLAMCH( 'S' )
          BIGNUM = ONE / SMLNUM
          CALL DLABAD( SMLNUM, BIGNUM )
          SMLNUM = SQRT( SMLNUM ) / EPS
          BIGNUM = ONE / SMLNUM
    *
    *     Scale A if max element outside range [SMLNUM,BIGNUM]
    *
          ANRM = ZLANGE( '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 ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
    *
    *     Permute the matrix to make it more nearly triangular
    *     (CWorkspace: none)
    *     (RWorkspace: need N)
    *
          IBAL = 1
          CALL ZGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
    *
    *     Reduce to upper Hessenberg form
    *     (CWorkspace: need 2*N, prefer N+N*NB)
    *     (RWorkspace: none)
    *
          ITAU = 1
          IWRK = N + ITAU
          CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
         $             LWORK-IWRK+1, IERR )
    *
          IF( WANTVS ) THEN
    *
    *        Copy Householder vectors to VS
    *
             CALL ZLACPY( 'L', N, N, A, LDA, VS, LDVS )
    *
    *        Generate unitary matrix in VS
    *        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
    *        (RWorkspace: none)
    *
             CALL ZUNGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
         $                LWORK-IWRK+1, IERR )
          END IF
    *
          SDIM = 0
    *
    *     Perform QR iteration, accumulating Schur vectors in VS if desired
    *     (CWorkspace: need 1, prefer HSWORK (see comments) )
    *     (RWorkspace: none)
    *
          IWRK = ITAU
          CALL ZHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, VS, LDVS,
         $             WORK( IWRK ), LWORK-IWRK+1, IEVAL )
          IF( IEVAL.GT.0 )
         $   INFO = IEVAL
    *
    *     Sort eigenvalues if desired
    *
          IF( WANTST .AND. INFO.EQ.0 ) THEN
             IF( SCALEA )
         $      CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR )
             DO 10 I = 1, N
                BWORK( I ) = SELECT( W( I ) )
       10    CONTINUE
    *
    *        Reorder eigenvalues and transform Schur vectors
    *        (CWorkspace: none)
    *        (RWorkspace: none)
    *
             CALL ZTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
         $                S, SEP, WORK( IWRK ), LWORK-IWRK+1, ICOND )
          END IF
    *
          IF( WANTVS ) THEN
    *
    *        Undo balancing
    *        (CWorkspace: none)
    *        (RWorkspace: need N)
    *
             CALL ZGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
         $                IERR )
          END IF
    *
          IF( SCALEA ) THEN
    *
    *        Undo scaling for the Schur form of A
    *
             CALL ZLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
             CALL ZCOPY( N, A, LDA+1, W, 1 )
          END IF
    *
          WORK( 1 ) = MAXWRK
          RETURN
    *
    *     End of ZGEES
    *
          END