zlaqr0.f

          SUBROUTINE ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
         $                   IHIZ, Z, LDZ, WORK, LWORK, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
          LOGICAL            WANTT, WANTZ
    *     ..
    *     .. Array Arguments ..
          COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
    *     ..
    *
    *     Purpose
    *     =======
    *
    *     ZLAQR0 computes the eigenvalues of a Hessenberg matrix H
    *     and, optionally, the matrices T and Z from the Schur decomposition
    *     H = Z T Z**H, where T is an upper triangular matrix (the
    *     Schur form), and Z is the unitary matrix of Schur vectors.
    *
    *     Optionally Z may be postmultiplied into an input unitary
    *     matrix Q so that this routine can give the Schur factorization
    *     of a matrix A which has been reduced to the Hessenberg form H
    *     by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
    *
    *     Arguments
    *     =========
    *
    *     WANTT   (input) LOGICAL
    *          = .TRUE. : the full Schur form T is required;
    *          = .FALSE.: only eigenvalues are required.
    *
    *     WANTZ   (input) LOGICAL
    *          = .TRUE. : the matrix of Schur vectors Z is required;
    *          = .FALSE.: Schur vectors are not required.
    *
    *     N     (input) INTEGER
    *           The order of the matrix H.  N .GE. 0.
    *
    *     ILO   (input) INTEGER
    *     IHI   (input) INTEGER
    *           It is assumed that H is already upper triangular in rows
    *           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
    *           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
    *           previous call to ZGEBAL, and then passed to ZGEHRD when the
    *           matrix output by ZGEBAL is reduced to Hessenberg form.
    *           Otherwise, ILO and IHI should be set to 1 and N,
    *           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
    *           If N = 0, then ILO = 1 and IHI = 0.
    *
    *     H     (input/output) COMPLEX*16 array, dimension (LDH,N)
    *           On entry, the upper Hessenberg matrix H.
    *           On exit, if INFO = 0 and WANTT is .TRUE., then H
    *           contains the upper triangular matrix T from the Schur
    *           decomposition (the Schur form). If INFO = 0 and WANT is
    *           .FALSE., then the contents of H are unspecified on exit.
    *           (The output value of H when INFO.GT.0 is given under the
    *           description of INFO below.)
    *
    *           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
    *           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
    *
    *     LDH   (input) INTEGER
    *           The leading dimension of the array H. LDH .GE. max(1,N).
    *
    *     W        (output) COMPLEX*16 array, dimension (N)
    *           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
    *           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
    *           stored in the same order as on the diagonal of the Schur
    *           form returned in H, with W(i) = H(i,i).
    *
    *     Z     (input/output) COMPLEX*16 array, dimension (LDZ,IHI)
    *           If WANTZ is .FALSE., then Z is not referenced.
    *           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
    *           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
    *           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
    *           (The output value of Z when INFO.GT.0 is given under
    *           the description of INFO below.)
    *
    *     LDZ   (input) INTEGER
    *           The leading dimension of the array Z.  if WANTZ is .TRUE.
    *           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
    *
    *     WORK  (workspace/output) COMPLEX*16 array, dimension LWORK
    *           On exit, if LWORK = -1, WORK(1) returns an estimate of
    *           the optimal value for LWORK.
    *
    *     LWORK (input) INTEGER
    *           The dimension of the array WORK.  LWORK .GE. max(1,N)
    *           is sufficient, but LWORK typically as large as 6*N may
    *           be required for optimal performance.  A workspace query
    *           to determine the optimal workspace size is recommended.
    *
    *           If LWORK = -1, then ZLAQR0 does a workspace query.
    *           In this case, ZLAQR0 checks the input parameters and
    *           estimates the optimal workspace size for the given
    *           values of N, ILO and IHI.  The estimate is returned
    *           in WORK(1).  No error message related to LWORK is
    *           issued by XERBLA.  Neither H nor Z are accessed.
    *
    *
    *     INFO  (output) INTEGER
    *             =  0:  successful exit
    *           .GT. 0:  if INFO = i, ZLAQR0 failed to compute all of
    *                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
    *                and WI contain those eigenvalues which have been
    *                successfully computed.  (Failures are rare.)
    *
    *                If INFO .GT. 0 and WANT is .FALSE., then on exit,
    *                the remaining unconverged eigenvalues are the eigen-
    *                values of the upper Hessenberg matrix rows and
    *                columns ILO through INFO of the final, output
    *                value of H.
    *
    *                If INFO .GT. 0 and WANTT is .TRUE., then on exit
    *
    *           (*)  (initial value of H)*U  = U*(final value of H)
    *
    *                where U is a unitary matrix.  The final
    *                value of  H is upper Hessenberg and triangular in
    *                rows and columns INFO+1 through IHI.
    *
    *                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
    *
    *                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
    *                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
    *
    *                where U is the unitary matrix in (*) (regard-
    *                less of the value of WANTT.)
    *
    *                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
    *                accessed.
    *
    *     ================================================================
    *     Based on contributions by
    *        Karen Braman and Ralph Byers, Department of Mathematics,
    *        University of Kansas, USA
    *
    *     ================================================================
    *     References:
    *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
    *       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
    *       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
    *       929--947, 2002.
    *
    *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
    *       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
    *       of Matrix Analysis, volume 23, pages 948--973, 2002.
    *
    *     ================================================================
    *     .. Parameters ..
    *
    *     ==== Matrices of order NTINY or smaller must be processed by
    *     .    ZLAHQR because of insufficient subdiagonal scratch space.
    *     .    (This is a hard limit.) ====
    *
    *     ==== Exceptional deflation windows:  try to cure rare
    *     .    slow convergence by increasing the size of the
    *     .    deflation window after KEXNW iterations. =====
    *
    *     ==== Exceptional shifts: try to cure rare slow convergence
    *     .    with ad-hoc exceptional shifts every KEXSH iterations.
    *     .    The constants WILK1 and WILK2 are used to form the
    *     .    exceptional shifts. ====
    *
          INTEGER            NTINY
          PARAMETER          ( NTINY = 11 )
          INTEGER            KEXNW, KEXSH
          PARAMETER          ( KEXNW = 5, KEXSH = 6 )
          DOUBLE PRECISION   WILK1
          PARAMETER          ( WILK1 = 0.75d0 )
          COMPLEX*16         ZERO, ONE
          PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ),
         $                   ONE = ( 1.0d0, 0.0d0 ) )
          DOUBLE PRECISION   TWO
          PARAMETER          ( TWO = 2.0d0 )
    *     ..
    *     .. Local Scalars ..
          COMPLEX*16         AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
          DOUBLE PRECISION   S
          INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
         $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
         $                   LWKOPT, NDFL, NH, NHO, NIBBLE, NMIN, NS, NSMAX,
         $                   NSR, NVE, NW, NWMAX, NWR
          LOGICAL            NWINC, SORTED
          CHARACTER          JBCMPZ*2
    *     ..
    *     .. External Functions ..
          INTEGER            ILAENV
          EXTERNAL           ILAENV
    *     ..
    *     .. Local Arrays ..
          COMPLEX*16         ZDUM( 1, 1 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           ZLACPY, ZLAHQR, ZLAQR3, ZLAQR4, ZLAQR5
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD,
         $                   SQRT
    *     ..
    *     .. Statement Functions ..
          DOUBLE PRECISION   CABS1
    *     ..
    *     .. Statement Function definitions ..
          CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
    *     ..
    *     .. Executable Statements ..
          INFO = 0
    *
    *     ==== Quick return for N = 0: nothing to do. ====
    *
          IF( N.EQ.0 ) THEN
             WORK( 1 ) = ONE
             RETURN
          END IF
    *
    *     ==== Set up job flags for ILAENV. ====
    *
          IF( WANTT ) THEN
             JBCMPZ( 1: 1 ) = 'S'
          ELSE
             JBCMPZ( 1: 1 ) = 'E'
          END IF
          IF( WANTZ ) THEN
             JBCMPZ( 2: 2 ) = 'V'
          ELSE
             JBCMPZ( 2: 2 ) = 'N'
          END IF
    *
    *     ==== Tiny matrices must use ZLAHQR. ====
    *
          IF( N.LE.NTINY ) THEN
    *
    *        ==== Estimate optimal workspace. ====
    *
             LWKOPT = 1
             IF( LWORK.NE.-1 )
         $      CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
         $                   IHIZ, Z, LDZ, INFO )
          ELSE
    *
    *        ==== Use small bulge multi-shift QR with aggressive early
    *        .    deflation on larger-than-tiny matrices. ====
    *
    *        ==== Hope for the best. ====
    *
             INFO = 0
    *
    *        ==== NWR = recommended deflation window size.  At this
    *        .    point,  N .GT. NTINY = 11, so there is enough
    *        .    subdiagonal workspace for NWR.GE.2 as required.
    *        .    (In fact, there is enough subdiagonal space for
    *        .    NWR.GE.3.) ====
    *
             NWR = ILAENV( 13, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
             NWR = MAX( 2, NWR )
             NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
             NW = NWR
    *
    *        ==== NSR = recommended number of simultaneous shifts.
    *        .    At this point N .GT. NTINY = 11, so there is at
    *        .    enough subdiagonal workspace for NSR to be even
    *        .    and greater than or equal to two as required. ====
    *
             NSR = ILAENV( 15, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
             NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
             NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
    *
    *        ==== Estimate optimal workspace ====
    *
    *        ==== Workspace query call to ZLAQR3 ====
    *
             CALL ZLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
         $                IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
         $                LDH, WORK, -1 )
    *
    *        ==== Optimal workspace = MAX(ZLAQR5, ZLAQR3) ====
    *
             LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
    *
    *        ==== Quick return in case of workspace query. ====
    *
             IF( LWORK.EQ.-1 ) THEN
                WORK( 1 ) = DCMPLX( LWKOPT, 0 )
                RETURN
             END IF
    *
    *        ==== ZLAHQR/ZLAQR0 crossover point ====
    *
             NMIN = ILAENV( 12, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
             NMIN = MAX( NTINY, NMIN )
    *
    *        ==== Nibble crossover point ====
    *
             NIBBLE = ILAENV( 14, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
             NIBBLE = MAX( 0, NIBBLE )
    *
    *        ==== Accumulate reflections during ttswp?  Use block
    *        .    2-by-2 structure during matrix-matrix multiply? ====
    *
             KACC22 = ILAENV( 16, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
             KACC22 = MAX( 0, KACC22 )
             KACC22 = MIN( 2, KACC22 )
    *
    *        ==== NWMAX = the largest possible deflation window for
    *        .    which there is sufficient workspace. ====
    *
             NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
    *
    *        ==== NSMAX = the Largest number of simultaneous shifts
    *        .    for which there is sufficient workspace. ====
    *
             NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
             NSMAX = NSMAX - MOD( NSMAX, 2 )
    *
    *        ==== NDFL: an iteration count restarted at deflation. ====
    *
             NDFL = 1
    *
    *        ==== ITMAX = iteration limit ====
    *
             ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
    *
    *        ==== Last row and column in the active block ====
    *
             KBOT = IHI
    *
    *        ==== Main Loop ====
    *
             DO 70 IT = 1, ITMAX
    *
    *           ==== Done when KBOT falls below ILO ====
    *
                IF( KBOT.LT.ILO )
         $         GO TO 80
    *
    *           ==== Locate active block ====
    *
                DO 10 K = KBOT, ILO + 1, -1
                   IF( H( K, K-1 ).EQ.ZERO )
         $            GO TO 20
       10       CONTINUE
                K = ILO
       20       CONTINUE
                KTOP = K
    *
    *           ==== Select deflation window size ====
    *
                NH = KBOT - KTOP + 1
                IF( NDFL.LT.KEXNW .OR. NH.LT.NW ) THEN
    *
    *              ==== Typical deflation window.  If possible and
    *              .    advisable, nibble the entire active block.
    *              .    If not, use size NWR or NWR+1 depending upon
    *              .    which has the smaller corresponding subdiagonal
    *              .    entry (a heuristic). ====
    *
                   NWINC = .TRUE.
                   IF( NH.LE.MIN( NMIN, NWMAX ) ) THEN
                      NW = NH
                   ELSE
                      NW = MIN( NWR, NH, NWMAX )
                      IF( NW.LT.NWMAX ) THEN
                         IF( NW.GE.NH-1 ) THEN
                            NW = NH
                         ELSE
                            KWTOP = KBOT - NW + 1
                            IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
         $                      CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
                         END IF
                      END IF
                   END IF
                ELSE
    *
    *              ==== Exceptional deflation window.  If there have
    *              .    been no deflations in KEXNW or more iterations,
    *              .    then vary the deflation window size.   At first,
    *              .    because, larger windows are, in general, more
    *              .    powerful than smaller ones, rapidly increase the
    *              .    window up to the maximum reasonable and possible.
    *              .    Then maybe try a slightly smaller window.  ====
    *
                   IF( NWINC .AND. NW.LT.MIN( NWMAX, NH ) ) THEN
                      NW = MIN( NWMAX, NH, 2*NW )
                   ELSE
                      NWINC = .FALSE.
                      IF( NW.EQ.NH .AND. NH.GT.2 )
         $               NW = NH - 1
                   END IF
                END IF
    *
    *           ==== Aggressive early deflation:
    *           .    split workspace under the subdiagonal into
    *           .      - an nw-by-nw work array V in the lower
    *           .        left-hand-corner,
    *           .      - an NW-by-at-least-NW-but-more-is-better
    *           .        (NW-by-NHO) horizontal work array along
    *           .        the bottom edge,
    *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
    *           .        vertical work array along the left-hand-edge.
    *           .        ====
    *
                KV = N - NW + 1
                KT = NW + 1
                NHO = ( N-NW-1 ) - KT + 1
                KWV = NW + 2
                NVE = ( N-NW ) - KWV + 1
    *
    *           ==== Aggressive early deflation ====
    *
                CALL ZLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
         $                   IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
         $                   H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
         $                   LWORK )
    *
    *           ==== Adjust KBOT accounting for new deflations. ====
    *
                KBOT = KBOT - LD
    *
    *           ==== KS points to the shifts. ====
    *
                KS = KBOT - LS + 1
    *
    *           ==== Skip an expensive QR sweep if there is a (partly
    *           .    heuristic) reason to expect that many eigenvalues
    *           .    will deflate without it.  Here, the QR sweep is
    *           .    skipped if many eigenvalues have just been deflated
    *           .    or if the remaining active block is small.
    *
                IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
         $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
    *
    *              ==== NS = nominal number of simultaneous shifts.
    *              .    This may be lowered (slightly) if ZLAQR3
    *              .    did not provide that many shifts. ====
    *
                   NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
                   NS = NS - MOD( NS, 2 )
    *
    *              ==== If there have been no deflations
    *              .    in a multiple of KEXSH iterations,
    *              .    then try exceptional shifts.
    *              .    Otherwise use shifts provided by
    *              .    ZLAQR3 above or from the eigenvalues
    *              .    of a trailing principal submatrix. ====
    *
                   IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
                      KS = KBOT - NS + 1
                      DO 30 I = KBOT, KS + 1, -2
                         W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
                         W( I-1 ) = W( I )
       30             CONTINUE
                   ELSE
    *
    *                 ==== Got NS/2 or fewer shifts? Use ZLAQR4 or
    *                 .    ZLAHQR on a trailing principal submatrix to
    *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
    *                 .    there is enough space below the subdiagonal
    *                 .    to fit an NS-by-NS scratch array.) ====
    *
                      IF( KBOT-KS+1.LE.NS / 2 ) THEN
                         KS = KBOT - NS + 1
                         KT = N - NS + 1
                         CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH,
         $                            H( KT, 1 ), LDH )
                         IF( NS.GT.NMIN ) THEN
                            CALL ZLAQR4( .false., .false., NS, 1, NS,
         $                               H( KT, 1 ), LDH, W( KS ), 1, 1,
         $                               ZDUM, 1, WORK, LWORK, INF )
                         ELSE
                            CALL ZLAHQR( .false., .false., NS, 1, NS,
         $                               H( KT, 1 ), LDH, W( KS ), 1, 1,
         $                               ZDUM, 1, INF )
                         END IF
                         KS = KS + INF
    *
    *                    ==== In case of a rare QR failure use
    *                    .    eigenvalues of the trailing 2-by-2
    *                    .    principal submatrix.  Scale to avoid
    *                    .    overflows, underflows and subnormals.
    *                    .    (The scale factor S can not be zero,
    *                    .    because H(KBOT,KBOT-1) is nonzero.) ====
    *
                         IF( KS.GE.KBOT ) THEN
                            S = CABS1( H( KBOT-1, KBOT-1 ) ) +
         $                      CABS1( H( KBOT, KBOT-1 ) ) +
         $                      CABS1( H( KBOT-1, KBOT ) ) +
         $                      CABS1( H( KBOT, KBOT ) )
                            AA = H( KBOT-1, KBOT-1 ) / S
                            CC = H( KBOT, KBOT-1 ) / S
                            BB = H( KBOT-1, KBOT ) / S
                            DD = H( KBOT, KBOT ) / S
                            TR2 = ( AA+DD ) / TWO
                            DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
                            RTDISC = SQRT( -DET )
                            W( KBOT-1 ) = ( TR2+RTDISC )*S
                            W( KBOT ) = ( TR2-RTDISC )*S
    *
                            KS = KBOT - 1
                         END IF
                      END IF
    *
                      IF( KBOT-KS+1.GT.NS ) THEN
    *
    *                    ==== Sort the shifts (Helps a little) ====
    *
                         SORTED = .false.
                         DO 50 K = KBOT, KS + 1, -1
                            IF( SORTED )
         $                     GO TO 60
                            SORTED = .true.
                            DO 40 I = KS, K - 1
                               IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
         $                          THEN
                                  SORTED = .false.
                                  SWAP = W( I )
                                  W( I ) = W( I+1 )
                                  W( I+1 ) = SWAP
                               END IF
       40                   CONTINUE
       50                CONTINUE
       60                CONTINUE
                      END IF
                   END IF
    *
    *              ==== If there are only two shifts, then use
    *              .    only one.  ====
    *
                   IF( KBOT-KS+1.EQ.2 ) THEN
                      IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
         $                CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
                         W( KBOT-1 ) = W( KBOT )
                      ELSE
                         W( KBOT ) = W( KBOT-1 )
                      END IF
                   END IF
    *
    *              ==== Use up to NS of the the smallest magnatiude
    *              .    shifts.  If there aren't NS shifts available,
    *              .    then use them all, possibly dropping one to
    *              .    make the number of shifts even. ====
    *
                   NS = MIN( NS, KBOT-KS+1 )
                   NS = NS - MOD( NS, 2 )
                   KS = KBOT - NS + 1
    *
    *              ==== Small-bulge multi-shift QR sweep:
    *              .    split workspace under the subdiagonal into
    *              .    - a KDU-by-KDU work array U in the lower
    *              .      left-hand-corner,
    *              .    - a KDU-by-at-least-KDU-but-more-is-better
    *              .      (KDU-by-NHo) horizontal work array WH along
    *              .      the bottom edge,
    *              .    - and an at-least-KDU-but-more-is-better-by-KDU
    *              .      (NVE-by-KDU) vertical work WV arrow along
    *              .      the left-hand-edge. ====
    *
                   KDU = 3*NS - 3
                   KU = N - KDU + 1
                   KWH = KDU + 1
                   NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
                   KWV = KDU + 4
                   NVE = N - KDU - KWV + 1
    *
    *              ==== Small-bulge multi-shift QR sweep ====
    *
                   CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
         $                      W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
         $                      3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
         $                      NHO, H( KU, KWH ), LDH )
                END IF
    *
    *           ==== Note progress (or the lack of it). ====
    *
                IF( LD.GT.0 ) THEN
                   NDFL = 1
                ELSE
                   NDFL = NDFL + 1
                END IF
    *
    *           ==== End of main loop ====
       70    CONTINUE
    *
    *        ==== Iteration limit exceeded.  Set INFO to show where
    *        .    the problem occurred and exit. ====
    *
             INFO = KBOT
       80    CONTINUE
          END IF
    *
    *     ==== Return the optimal value of LWORK. ====
    *
          WORK( 1 ) = DCMPLX( LWKOPT, 0 )
    *
    *     ==== End of ZLAQR0 ====
    *
          END