LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zlaqr4.f
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00001 *> \brief \b ZLAQR4
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download ZLAQR4 + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr4.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr4.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
00022 *                          IHIZ, Z, LDZ, WORK, LWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
00026 *       LOGICAL            WANTT, WANTZ
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *>    ZLAQR4 implements one level of recursion for ZLAQR0.
00039 *>    It is a complete implementation of the small bulge multi-shift
00040 *>    QR algorithm.  It may be called by ZLAQR0 and, for large enough
00041 *>    deflation window size, it may be called by ZLAQR3.  This
00042 *>    subroutine is identical to ZLAQR0 except that it calls ZLAQR2
00043 *>    instead of ZLAQR3.
00044 *>
00045 *>    ZLAQR4 computes the eigenvalues of a Hessenberg matrix H
00046 *>    and, optionally, the matrices T and Z from the Schur decomposition
00047 *>    H = Z T Z**H, where T is an upper triangular matrix (the
00048 *>    Schur form), and Z is the unitary matrix of Schur vectors.
00049 *>
00050 *>    Optionally Z may be postmultiplied into an input unitary
00051 *>    matrix Q so that this routine can give the Schur factorization
00052 *>    of a matrix A which has been reduced to the Hessenberg form H
00053 *>    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
00054 *> \endverbatim
00055 *
00056 *  Arguments:
00057 *  ==========
00058 *
00059 *> \param[in] WANTT
00060 *> \verbatim
00061 *>          WANTT is LOGICAL
00062 *>          = .TRUE. : the full Schur form T is required;
00063 *>          = .FALSE.: only eigenvalues are required.
00064 *> \endverbatim
00065 *>
00066 *> \param[in] WANTZ
00067 *> \verbatim
00068 *>          WANTZ is LOGICAL
00069 *>          = .TRUE. : the matrix of Schur vectors Z is required;
00070 *>          = .FALSE.: Schur vectors are not required.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] N
00074 *> \verbatim
00075 *>          N is INTEGER
00076 *>           The order of the matrix H.  N .GE. 0.
00077 *> \endverbatim
00078 *>
00079 *> \param[in] ILO
00080 *> \verbatim
00081 *>          ILO is INTEGER
00082 *> \endverbatim
00083 *>
00084 *> \param[in] IHI
00085 *> \verbatim
00086 *>          IHI is INTEGER
00087 *>           It is assumed that H is already upper triangular in rows
00088 *>           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
00089 *>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
00090 *>           previous call to ZGEBAL, and then passed to ZGEHRD when the
00091 *>           matrix output by ZGEBAL is reduced to Hessenberg form.
00092 *>           Otherwise, ILO and IHI should be set to 1 and N,
00093 *>           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
00094 *>           If N = 0, then ILO = 1 and IHI = 0.
00095 *> \endverbatim
00096 *>
00097 *> \param[in,out] H
00098 *> \verbatim
00099 *>          H is COMPLEX*16 array, dimension (LDH,N)
00100 *>           On entry, the upper Hessenberg matrix H.
00101 *>           On exit, if INFO = 0 and WANTT is .TRUE., then H
00102 *>           contains the upper triangular matrix T from the Schur
00103 *>           decomposition (the Schur form). If INFO = 0 and WANT is
00104 *>           .FALSE., then the contents of H are unspecified on exit.
00105 *>           (The output value of H when INFO.GT.0 is given under the
00106 *>           description of INFO below.)
00107 *>
00108 *>           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
00109 *>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
00110 *> \endverbatim
00111 *>
00112 *> \param[in] LDH
00113 *> \verbatim
00114 *>          LDH is INTEGER
00115 *>           The leading dimension of the array H. LDH .GE. max(1,N).
00116 *> \endverbatim
00117 *>
00118 *> \param[out] W
00119 *> \verbatim
00120 *>          W is COMPLEX*16 array, dimension (N)
00121 *>           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
00122 *>           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
00123 *>           stored in the same order as on the diagonal of the Schur
00124 *>           form returned in H, with W(i) = H(i,i).
00125 *> \endverbatim
00126 *>
00127 *> \param[in] ILOZ
00128 *> \verbatim
00129 *>          ILOZ is INTEGER
00130 *> \endverbatim
00131 *>
00132 *> \param[in] IHIZ
00133 *> \verbatim
00134 *>          IHIZ is INTEGER
00135 *>           Specify the rows of Z to which transformations must be
00136 *>           applied if WANTZ is .TRUE..
00137 *>           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
00138 *> \endverbatim
00139 *>
00140 *> \param[in,out] Z
00141 *> \verbatim
00142 *>          Z is COMPLEX*16 array, dimension (LDZ,IHI)
00143 *>           If WANTZ is .FALSE., then Z is not referenced.
00144 *>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
00145 *>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
00146 *>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
00147 *>           (The output value of Z when INFO.GT.0 is given under
00148 *>           the description of INFO below.)
00149 *> \endverbatim
00150 *>
00151 *> \param[in] LDZ
00152 *> \verbatim
00153 *>          LDZ is INTEGER
00154 *>           The leading dimension of the array Z.  if WANTZ is .TRUE.
00155 *>           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
00156 *> \endverbatim
00157 *>
00158 *> \param[out] WORK
00159 *> \verbatim
00160 *>          WORK is COMPLEX*16 array, dimension LWORK
00161 *>           On exit, if LWORK = -1, WORK(1) returns an estimate of
00162 *>           the optimal value for LWORK.
00163 *> \endverbatim
00164 *>
00165 *> \param[in] LWORK
00166 *> \verbatim
00167 *>          LWORK is INTEGER
00168 *>           The dimension of the array WORK.  LWORK .GE. max(1,N)
00169 *>           is sufficient, but LWORK typically as large as 6*N may
00170 *>           be required for optimal performance.  A workspace query
00171 *>           to determine the optimal workspace size is recommended.
00172 *>
00173 *>           If LWORK = -1, then ZLAQR4 does a workspace query.
00174 *>           In this case, ZLAQR4 checks the input parameters and
00175 *>           estimates the optimal workspace size for the given
00176 *>           values of N, ILO and IHI.  The estimate is returned
00177 *>           in WORK(1).  No error message related to LWORK is
00178 *>           issued by XERBLA.  Neither H nor Z are accessed.
00179 *> \endverbatim
00180 *>
00181 *> \param[out] INFO
00182 *> \verbatim
00183 *>          INFO is INTEGER
00184 *>             =  0:  successful exit
00185 *>           .GT. 0:  if INFO = i, ZLAQR4 failed to compute all of
00186 *>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00187 *>                and WI contain those eigenvalues which have been
00188 *>                successfully computed.  (Failures are rare.)
00189 *>
00190 *>                If INFO .GT. 0 and WANT is .FALSE., then on exit,
00191 *>                the remaining unconverged eigenvalues are the eigen-
00192 *>                values of the upper Hessenberg matrix rows and
00193 *>                columns ILO through INFO of the final, output
00194 *>                value of H.
00195 *>
00196 *>                If INFO .GT. 0 and WANTT is .TRUE., then on exit
00197 *>
00198 *>           (*)  (initial value of H)*U  = U*(final value of H)
00199 *>
00200 *>                where U is a unitary matrix.  The final
00201 *>                value of  H is upper Hessenberg and triangular in
00202 *>                rows and columns INFO+1 through IHI.
00203 *>
00204 *>                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
00205 *>
00206 *>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
00207 *>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
00208 *>
00209 *>                where U is the unitary matrix in (*) (regard-
00210 *>                less of the value of WANTT.)
00211 *>
00212 *>                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
00213 *>                accessed.
00214 *> \endverbatim
00215 *
00216 *  Authors:
00217 *  ========
00218 *
00219 *> \author Univ. of Tennessee 
00220 *> \author Univ. of California Berkeley 
00221 *> \author Univ. of Colorado Denver 
00222 *> \author NAG Ltd. 
00223 *
00224 *> \date November 2011
00225 *
00226 *> \ingroup complex16OTHERauxiliary
00227 *
00228 *> \par Contributors:
00229 *  ==================
00230 *>
00231 *>       Karen Braman and Ralph Byers, Department of Mathematics,
00232 *>       University of Kansas, USA
00233 *
00234 *> \par References:
00235 *  ================
00236 *>
00237 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00238 *>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00239 *>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00240 *>       929--947, 2002.
00241 *> \n
00242 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00243 *>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00244 *>       of Matrix Analysis, volume 23, pages 948--973, 2002.
00245 *>
00246 *  =====================================================================
00247       SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
00248      $                   IHIZ, Z, LDZ, WORK, LWORK, INFO )
00249 *
00250 *  -- LAPACK auxiliary routine (version 3.4.0) --
00251 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00252 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00253 *     November 2011
00254 *
00255 *     .. Scalar Arguments ..
00256       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
00257       LOGICAL            WANTT, WANTZ
00258 *     ..
00259 *     .. Array Arguments ..
00260       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
00261 *     ..
00262 *
00263 *  ================================================================
00264 *
00265 *     .. Parameters ..
00266 *
00267 *     ==== Matrices of order NTINY or smaller must be processed by
00268 *     .    ZLAHQR because of insufficient subdiagonal scratch space.
00269 *     .    (This is a hard limit.) ====
00270       INTEGER            NTINY
00271       PARAMETER          ( NTINY = 11 )
00272 *
00273 *     ==== Exceptional deflation windows:  try to cure rare
00274 *     .    slow convergence by varying the size of the
00275 *     .    deflation window after KEXNW iterations. ====
00276       INTEGER            KEXNW
00277       PARAMETER          ( KEXNW = 5 )
00278 *
00279 *     ==== Exceptional shifts: try to cure rare slow convergence
00280 *     .    with ad-hoc exceptional shifts every KEXSH iterations.
00281 *     .    ====
00282       INTEGER            KEXSH
00283       PARAMETER          ( KEXSH = 6 )
00284 *
00285 *     ==== The constant WILK1 is used to form the exceptional
00286 *     .    shifts. ====
00287       DOUBLE PRECISION   WILK1
00288       PARAMETER          ( WILK1 = 0.75d0 )
00289       COMPLEX*16         ZERO, ONE
00290       PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ),
00291      $                   ONE = ( 1.0d0, 0.0d0 ) )
00292       DOUBLE PRECISION   TWO
00293       PARAMETER          ( TWO = 2.0d0 )
00294 *     ..
00295 *     .. Local Scalars ..
00296       COMPLEX*16         AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
00297       DOUBLE PRECISION   S
00298       INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
00299      $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
00300      $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
00301      $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
00302       LOGICAL            SORTED
00303       CHARACTER          JBCMPZ*2
00304 *     ..
00305 *     .. External Functions ..
00306       INTEGER            ILAENV
00307       EXTERNAL           ILAENV
00308 *     ..
00309 *     .. Local Arrays ..
00310       COMPLEX*16         ZDUM( 1, 1 )
00311 *     ..
00312 *     .. External Subroutines ..
00313       EXTERNAL           ZLACPY, ZLAHQR, ZLAQR2, ZLAQR5
00314 *     ..
00315 *     .. Intrinsic Functions ..
00316       INTRINSIC          ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD,
00317      $                   SQRT
00318 *     ..
00319 *     .. Statement Functions ..
00320       DOUBLE PRECISION   CABS1
00321 *     ..
00322 *     .. Statement Function definitions ..
00323       CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
00324 *     ..
00325 *     .. Executable Statements ..
00326       INFO = 0
00327 *
00328 *     ==== Quick return for N = 0: nothing to do. ====
00329 *
00330       IF( N.EQ.0 ) THEN
00331          WORK( 1 ) = ONE
00332          RETURN
00333       END IF
00334 *
00335       IF( N.LE.NTINY ) THEN
00336 *
00337 *        ==== Tiny matrices must use ZLAHQR. ====
00338 *
00339          LWKOPT = 1
00340          IF( LWORK.NE.-1 )
00341      $      CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
00342      $                   IHIZ, Z, LDZ, INFO )
00343       ELSE
00344 *
00345 *        ==== Use small bulge multi-shift QR with aggressive early
00346 *        .    deflation on larger-than-tiny matrices. ====
00347 *
00348 *        ==== Hope for the best. ====
00349 *
00350          INFO = 0
00351 *
00352 *        ==== Set up job flags for ILAENV. ====
00353 *
00354          IF( WANTT ) THEN
00355             JBCMPZ( 1: 1 ) = 'S'
00356          ELSE
00357             JBCMPZ( 1: 1 ) = 'E'
00358          END IF
00359          IF( WANTZ ) THEN
00360             JBCMPZ( 2: 2 ) = 'V'
00361          ELSE
00362             JBCMPZ( 2: 2 ) = 'N'
00363          END IF
00364 *
00365 *        ==== NWR = recommended deflation window size.  At this
00366 *        .    point,  N .GT. NTINY = 11, so there is enough
00367 *        .    subdiagonal workspace for NWR.GE.2 as required.
00368 *        .    (In fact, there is enough subdiagonal space for
00369 *        .    NWR.GE.3.) ====
00370 *
00371          NWR = ILAENV( 13, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00372          NWR = MAX( 2, NWR )
00373          NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
00374 *
00375 *        ==== NSR = recommended number of simultaneous shifts.
00376 *        .    At this point N .GT. NTINY = 11, so there is at
00377 *        .    enough subdiagonal workspace for NSR to be even
00378 *        .    and greater than or equal to two as required. ====
00379 *
00380          NSR = ILAENV( 15, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00381          NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
00382          NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
00383 *
00384 *        ==== Estimate optimal workspace ====
00385 *
00386 *        ==== Workspace query call to ZLAQR2 ====
00387 *
00388          CALL ZLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
00389      $                IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
00390      $                LDH, WORK, -1 )
00391 *
00392 *        ==== Optimal workspace = MAX(ZLAQR5, ZLAQR2) ====
00393 *
00394          LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
00395 *
00396 *        ==== Quick return in case of workspace query. ====
00397 *
00398          IF( LWORK.EQ.-1 ) THEN
00399             WORK( 1 ) = DCMPLX( LWKOPT, 0 )
00400             RETURN
00401          END IF
00402 *
00403 *        ==== ZLAHQR/ZLAQR0 crossover point ====
00404 *
00405          NMIN = ILAENV( 12, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00406          NMIN = MAX( NTINY, NMIN )
00407 *
00408 *        ==== Nibble crossover point ====
00409 *
00410          NIBBLE = ILAENV( 14, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00411          NIBBLE = MAX( 0, NIBBLE )
00412 *
00413 *        ==== Accumulate reflections during ttswp?  Use block
00414 *        .    2-by-2 structure during matrix-matrix multiply? ====
00415 *
00416          KACC22 = ILAENV( 16, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00417          KACC22 = MAX( 0, KACC22 )
00418          KACC22 = MIN( 2, KACC22 )
00419 *
00420 *        ==== NWMAX = the largest possible deflation window for
00421 *        .    which there is sufficient workspace. ====
00422 *
00423          NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
00424          NW = NWMAX
00425 *
00426 *        ==== NSMAX = the Largest number of simultaneous shifts
00427 *        .    for which there is sufficient workspace. ====
00428 *
00429          NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
00430          NSMAX = NSMAX - MOD( NSMAX, 2 )
00431 *
00432 *        ==== NDFL: an iteration count restarted at deflation. ====
00433 *
00434          NDFL = 1
00435 *
00436 *        ==== ITMAX = iteration limit ====
00437 *
00438          ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
00439 *
00440 *        ==== Last row and column in the active block ====
00441 *
00442          KBOT = IHI
00443 *
00444 *        ==== Main Loop ====
00445 *
00446          DO 70 IT = 1, ITMAX
00447 *
00448 *           ==== Done when KBOT falls below ILO ====
00449 *
00450             IF( KBOT.LT.ILO )
00451      $         GO TO 80
00452 *
00453 *           ==== Locate active block ====
00454 *
00455             DO 10 K = KBOT, ILO + 1, -1
00456                IF( H( K, K-1 ).EQ.ZERO )
00457      $            GO TO 20
00458    10       CONTINUE
00459             K = ILO
00460    20       CONTINUE
00461             KTOP = K
00462 *
00463 *           ==== Select deflation window size:
00464 *           .    Typical Case:
00465 *           .      If possible and advisable, nibble the entire
00466 *           .      active block.  If not, use size MIN(NWR,NWMAX)
00467 *           .      or MIN(NWR+1,NWMAX) depending upon which has
00468 *           .      the smaller corresponding subdiagonal entry
00469 *           .      (a heuristic).
00470 *           .
00471 *           .    Exceptional Case:
00472 *           .      If there have been no deflations in KEXNW or
00473 *           .      more iterations, then vary the deflation window
00474 *           .      size.   At first, because, larger windows are,
00475 *           .      in general, more powerful than smaller ones,
00476 *           .      rapidly increase the window to the maximum possible.
00477 *           .      Then, gradually reduce the window size. ====
00478 *
00479             NH = KBOT - KTOP + 1
00480             NWUPBD = MIN( NH, NWMAX )
00481             IF( NDFL.LT.KEXNW ) THEN
00482                NW = MIN( NWUPBD, NWR )
00483             ELSE
00484                NW = MIN( NWUPBD, 2*NW )
00485             END IF
00486             IF( NW.LT.NWMAX ) THEN
00487                IF( NW.GE.NH-1 ) THEN
00488                   NW = NH
00489                ELSE
00490                   KWTOP = KBOT - NW + 1
00491                   IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
00492      $                CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
00493                END IF
00494             END IF
00495             IF( NDFL.LT.KEXNW ) THEN
00496                NDEC = -1
00497             ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
00498                NDEC = NDEC + 1
00499                IF( NW-NDEC.LT.2 )
00500      $            NDEC = 0
00501                NW = NW - NDEC
00502             END IF
00503 *
00504 *           ==== Aggressive early deflation:
00505 *           .    split workspace under the subdiagonal into
00506 *           .      - an nw-by-nw work array V in the lower
00507 *           .        left-hand-corner,
00508 *           .      - an NW-by-at-least-NW-but-more-is-better
00509 *           .        (NW-by-NHO) horizontal work array along
00510 *           .        the bottom edge,
00511 *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
00512 *           .        vertical work array along the left-hand-edge.
00513 *           .        ====
00514 *
00515             KV = N - NW + 1
00516             KT = NW + 1
00517             NHO = ( N-NW-1 ) - KT + 1
00518             KWV = NW + 2
00519             NVE = ( N-NW ) - KWV + 1
00520 *
00521 *           ==== Aggressive early deflation ====
00522 *
00523             CALL ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
00524      $                   IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
00525      $                   H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
00526      $                   LWORK )
00527 *
00528 *           ==== Adjust KBOT accounting for new deflations. ====
00529 *
00530             KBOT = KBOT - LD
00531 *
00532 *           ==== KS points to the shifts. ====
00533 *
00534             KS = KBOT - LS + 1
00535 *
00536 *           ==== Skip an expensive QR sweep if there is a (partly
00537 *           .    heuristic) reason to expect that many eigenvalues
00538 *           .    will deflate without it.  Here, the QR sweep is
00539 *           .    skipped if many eigenvalues have just been deflated
00540 *           .    or if the remaining active block is small.
00541 *
00542             IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
00543      $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
00544 *
00545 *              ==== NS = nominal number of simultaneous shifts.
00546 *              .    This may be lowered (slightly) if ZLAQR2
00547 *              .    did not provide that many shifts. ====
00548 *
00549                NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
00550                NS = NS - MOD( NS, 2 )
00551 *
00552 *              ==== If there have been no deflations
00553 *              .    in a multiple of KEXSH iterations,
00554 *              .    then try exceptional shifts.
00555 *              .    Otherwise use shifts provided by
00556 *              .    ZLAQR2 above or from the eigenvalues
00557 *              .    of a trailing principal submatrix. ====
00558 *
00559                IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
00560                   KS = KBOT - NS + 1
00561                   DO 30 I = KBOT, KS + 1, -2
00562                      W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
00563                      W( I-1 ) = W( I )
00564    30             CONTINUE
00565                ELSE
00566 *
00567 *                 ==== Got NS/2 or fewer shifts? Use ZLAHQR
00568 *                 .    on a trailing principal submatrix to
00569 *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
00570 *                 .    there is enough space below the subdiagonal
00571 *                 .    to fit an NS-by-NS scratch array.) ====
00572 *
00573                   IF( KBOT-KS+1.LE.NS / 2 ) THEN
00574                      KS = KBOT - NS + 1
00575                      KT = N - NS + 1
00576                      CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH,
00577      $                            H( KT, 1 ), LDH )
00578                      CALL ZLAHQR( .false., .false., NS, 1, NS,
00579      $                            H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
00580      $                            1, INF )
00581                      KS = KS + INF
00582 *
00583 *                    ==== In case of a rare QR failure use
00584 *                    .    eigenvalues of the trailing 2-by-2
00585 *                    .    principal submatrix.  Scale to avoid
00586 *                    .    overflows, underflows and subnormals.
00587 *                    .    (The scale factor S can not be zero,
00588 *                    .    because H(KBOT,KBOT-1) is nonzero.) ====
00589 *
00590                      IF( KS.GE.KBOT ) THEN
00591                         S = CABS1( H( KBOT-1, KBOT-1 ) ) +
00592      $                      CABS1( H( KBOT, KBOT-1 ) ) +
00593      $                      CABS1( H( KBOT-1, KBOT ) ) +
00594      $                      CABS1( H( KBOT, KBOT ) )
00595                         AA = H( KBOT-1, KBOT-1 ) / S
00596                         CC = H( KBOT, KBOT-1 ) / S
00597                         BB = H( KBOT-1, KBOT ) / S
00598                         DD = H( KBOT, KBOT ) / S
00599                         TR2 = ( AA+DD ) / TWO
00600                         DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
00601                         RTDISC = SQRT( -DET )
00602                         W( KBOT-1 ) = ( TR2+RTDISC )*S
00603                         W( KBOT ) = ( TR2-RTDISC )*S
00604 *
00605                         KS = KBOT - 1
00606                      END IF
00607                   END IF
00608 *
00609                   IF( KBOT-KS+1.GT.NS ) THEN
00610 *
00611 *                    ==== Sort the shifts (Helps a little) ====
00612 *
00613                      SORTED = .false.
00614                      DO 50 K = KBOT, KS + 1, -1
00615                         IF( SORTED )
00616      $                     GO TO 60
00617                         SORTED = .true.
00618                         DO 40 I = KS, K - 1
00619                            IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
00620      $                          THEN
00621                               SORTED = .false.
00622                               SWAP = W( I )
00623                               W( I ) = W( I+1 )
00624                               W( I+1 ) = SWAP
00625                            END IF
00626    40                   CONTINUE
00627    50                CONTINUE
00628    60                CONTINUE
00629                   END IF
00630                END IF
00631 *
00632 *              ==== If there are only two shifts, then use
00633 *              .    only one.  ====
00634 *
00635                IF( KBOT-KS+1.EQ.2 ) THEN
00636                   IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
00637      $                CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
00638                      W( KBOT-1 ) = W( KBOT )
00639                   ELSE
00640                      W( KBOT ) = W( KBOT-1 )
00641                   END IF
00642                END IF
00643 *
00644 *              ==== Use up to NS of the the smallest magnatiude
00645 *              .    shifts.  If there aren't NS shifts available,
00646 *              .    then use them all, possibly dropping one to
00647 *              .    make the number of shifts even. ====
00648 *
00649                NS = MIN( NS, KBOT-KS+1 )
00650                NS = NS - MOD( NS, 2 )
00651                KS = KBOT - NS + 1
00652 *
00653 *              ==== Small-bulge multi-shift QR sweep:
00654 *              .    split workspace under the subdiagonal into
00655 *              .    - a KDU-by-KDU work array U in the lower
00656 *              .      left-hand-corner,
00657 *              .    - a KDU-by-at-least-KDU-but-more-is-better
00658 *              .      (KDU-by-NHo) horizontal work array WH along
00659 *              .      the bottom edge,
00660 *              .    - and an at-least-KDU-but-more-is-better-by-KDU
00661 *              .      (NVE-by-KDU) vertical work WV arrow along
00662 *              .      the left-hand-edge. ====
00663 *
00664                KDU = 3*NS - 3
00665                KU = N - KDU + 1
00666                KWH = KDU + 1
00667                NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
00668                KWV = KDU + 4
00669                NVE = N - KDU - KWV + 1
00670 *
00671 *              ==== Small-bulge multi-shift QR sweep ====
00672 *
00673                CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
00674      $                      W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
00675      $                      3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
00676      $                      NHO, H( KU, KWH ), LDH )
00677             END IF
00678 *
00679 *           ==== Note progress (or the lack of it). ====
00680 *
00681             IF( LD.GT.0 ) THEN
00682                NDFL = 1
00683             ELSE
00684                NDFL = NDFL + 1
00685             END IF
00686 *
00687 *           ==== End of main loop ====
00688    70    CONTINUE
00689 *
00690 *        ==== Iteration limit exceeded.  Set INFO to show where
00691 *        .    the problem occurred and exit. ====
00692 *
00693          INFO = KBOT
00694    80    CONTINUE
00695       END IF
00696 *
00697 *     ==== Return the optimal value of LWORK. ====
00698 *
00699       WORK( 1 ) = DCMPLX( LWKOPT, 0 )
00700 *
00701 *     ==== End of ZLAQR4 ====
00702 *
00703       END
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