OpenBLAS/lapack-netlib/SRC/claqr4.f

*> \brief \b CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLAQR4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claqr4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claqr4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claqr4.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
*                          IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       COMPLEX            H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    CLAQR4 implements one level of recursion for CLAQR0.
*>    It is a complete implementation of the small bulge multi-shift
*>    QR algorithm.  It may be called by CLAQR0 and, for large enough
*>    deflation window size, it may be called by CLAQR3.  This
*>    subroutine is identical to CLAQR0 except that it calls CLAQR2
*>    instead of CLAQR3.
*>
*>    CLAQR4 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.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          WANTT is LOGICAL
*>          = .TRUE. : the full Schur form T is required;
*>          = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          = .TRUE. : the matrix of Schur vectors Z is required;
*>          = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>           The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*>          IHI is INTEGER
*>           It is assumed that H is already upper triangular in rows
*>           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
*>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
*>           previous call to CGEBAL, and then passed to CGEHRD when the
*>           matrix output by CGEBAL is reduced to Hessenberg form.
*>           Otherwise, ILO and IHI should be set to 1 and N,
*>           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
*>           If N = 0, then ILO = 1 and IHI = 0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          H is COMPLEX 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 > 0 is given under the
*>           description of INFO below.)
*>
*>           This subroutine may explicitly set H(i,j) = 0 for i > j and
*>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>           The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is COMPLEX 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).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>           Specify the rows of Z to which transformations must be
*>           applied if WANTZ is .TRUE..
*>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is COMPLEX 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 > 0 is given under
*>           the description of INFO below.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>           The leading dimension of the array Z.  if WANTZ is .TRUE.
*>           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension LWORK
*>           On exit, if LWORK = -1, WORK(1) returns an estimate of
*>           the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>           The dimension of the array WORK.  LWORK >= 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 CLAQR4 does a workspace query.
*>           In this case, CLAQR4 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>             = 0:  successful exit
*>             > 0:  if INFO = i, CLAQR4 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not
*>                accessed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup complexOTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*
*> \par 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.
*> \n
*>       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.
*>
*  =====================================================================
      SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
     $                   IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.7.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2017
*
*     .. Scalar Arguments ..
      INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      COMPLEX            H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*
*  ================================================================
*
*     .. Parameters ..
*
*     ==== Matrices of order NTINY or smaller must be processed by
*     .    CLAHQR because of insufficient subdiagonal scratch space.
*     .    (This is a hard limit.) ====
      INTEGER            NTINY
      PARAMETER          ( NTINY = 15 )
*
*     ==== Exceptional deflation windows:  try to cure rare
*     .    slow convergence by varying the size of the
*     .    deflation window after KEXNW iterations. ====
      INTEGER            KEXNW
      PARAMETER          ( KEXNW = 5 )
*
*     ==== Exceptional shifts: try to cure rare slow convergence
*     .    with ad-hoc exceptional shifts every KEXSH iterations.
*     .    ====
      INTEGER            KEXSH
      PARAMETER          ( KEXSH = 6 )
*
*     ==== The constant WILK1 is used to form the exceptional
*     .    shifts. ====
      REAL               WILK1
      PARAMETER          ( WILK1 = 0.75e0 )
      COMPLEX            ZERO, ONE
      PARAMETER          ( ZERO = ( 0.0e0, 0.0e0 ),
     $                   ONE = ( 1.0e0, 0.0e0 ) )
      REAL               TWO
      PARAMETER          ( TWO = 2.0e0 )
*     ..
*     .. Local Scalars ..
      COMPLEX            AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
      REAL               S
      INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
     $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
     $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
     $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
      LOGICAL            SORTED
      CHARACTER          JBCMPZ*2
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ILAENV
*     ..
*     .. Local Arrays ..
      COMPLEX            ZDUM( 1, 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           CLACPY, CLAHQR, CLAQR2, CLAQR5
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, AIMAG, CMPLX, INT, MAX, MIN, MOD, REAL,
     $                   SQRT
*     ..
*     .. Statement Functions ..
      REAL               CABS1
*     ..
*     .. Statement Function definitions ..
      CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
*     ..
*     .. Executable Statements ..
      INFO = 0
*
*     ==== Quick return for N = 0: nothing to do. ====
*
      IF( N.EQ.0 ) THEN
         WORK( 1 ) = ONE
         RETURN
      END IF
*
      IF( N.LE.NTINY ) THEN
*
*        ==== Tiny matrices must use CLAHQR. ====
*
         LWKOPT = 1
         IF( LWORK.NE.-1 )
     $      CALL CLAHQR( 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
*
*        ==== 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
*
*        ==== NWR = recommended deflation window size.  At this
*        .    point,  N .GT. NTINY = 15, so there is enough
*        .    subdiagonal workspace for NWR.GE.2 as required.
*        .    (In fact, there is enough subdiagonal space for
*        .    NWR.GE.4.) ====
*
         NWR = ILAENV( 13, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NWR = MAX( 2, NWR )
         NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
*        ==== NSR = recommended number of simultaneous shifts.
*        .    At this point N .GT. NTINY = 15, so there is at
*        .    enough subdiagonal workspace for NSR to be even
*        .    and greater than or equal to two as required. ====
*
         NSR = ILAENV( 15, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
         NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
*        ==== Estimate optimal workspace ====
*
*        ==== Workspace query call to CLAQR2 ====
*
         CALL CLAQR2( 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(CLAQR5, CLAQR2) ====
*
         LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
*        ==== Quick return in case of workspace query. ====
*
         IF( LWORK.EQ.-1 ) THEN
            WORK( 1 ) = CMPLX( LWKOPT, 0 )
            RETURN
         END IF
*
*        ==== CLAHQR/CLAQR0 crossover point ====
*
         NMIN = ILAENV( 12, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
         NMIN = MAX( NTINY, NMIN )
*
*        ==== Nibble crossover point ====
*
         NIBBLE = ILAENV( 14, 'CLAQR4', 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, 'CLAQR4', 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 )
         NW = NWMAX
*
*        ==== NSMAX = the Largest number of simultaneous shifts
*        .    for which there is sufficient workspace. ====
*
         NSMAX = MIN( ( N-3 ) / 6, 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:
*           .    Typical Case:
*           .      If possible and advisable, nibble the entire
*           .      active block.  If not, use size MIN(NWR,NWMAX)
*           .      or MIN(NWR+1,NWMAX) depending upon which has
*           .      the smaller corresponding subdiagonal entry
*           .      (a heuristic).
*           .
*           .    Exceptional Case:
*           .      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 to the maximum possible.
*           .      Then, gradually reduce the window size. ====
*
            NH = KBOT - KTOP + 1
            NWUPBD = MIN( NH, NWMAX )
            IF( NDFL.LT.KEXNW ) THEN
               NW = MIN( NWUPBD, NWR )
            ELSE
               NW = MIN( NWUPBD, 2*NW )
            END IF
            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
            IF( NDFL.LT.KEXNW ) THEN
               NDEC = -1
            ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
               NDEC = NDEC + 1
               IF( NW-NDEC.LT.2 )
     $            NDEC = 0
               NW = NW - NDEC
            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 CLAQR2( 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 CLAQR2
*              .    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
*              .    CLAQR2 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 CLAHQR
*                 .    on a trailing principal submatrix to
*                 .    get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
*                 .    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 CLACPY( 'A', NS, NS, H( KS, KS ), LDH,
     $                            H( KT, 1 ), LDH )
                     CALL CLAHQR( .false., .false., NS, 1, NS,
     $                            H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
     $                            1, INF )
                     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 magnitude
*              .    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 = 2*NS
               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 CLAQR5( 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 ) = CMPLX( LWKOPT, 0 )
*
*     ==== End of CLAQR4 ====
*
      END