LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
claqr4.f
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00001 *> \brief \b CLAQR4
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CLAQR4 + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claqr4.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CLAQR4( 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            H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
00030 *       ..
00031 *  
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *>    CLAQR4 implements one level of recursion for CLAQR0.
00040 *>    It is a complete implementation of the small bulge multi-shift
00041 *>    QR algorithm.  It may be called by CLAQR0 and, for large enough
00042 *>    deflation window size, it may be called by CLAQR3.  This
00043 *>    subroutine is identical to CLAQR0 except that it calls CLAQR2
00044 *>    instead of CLAQR3.
00045 *>
00046 *>    CLAQR4 computes the eigenvalues of a Hessenberg matrix H
00047 *>    and, optionally, the matrices T and Z from the Schur decomposition
00048 *>    H = Z T Z**H, where T is an upper triangular matrix (the
00049 *>    Schur form), and Z is the unitary matrix of Schur vectors.
00050 *>
00051 *>    Optionally Z may be postmultiplied into an input unitary
00052 *>    matrix Q so that this routine can give the Schur factorization
00053 *>    of a matrix A which has been reduced to the Hessenberg form H
00054 *>    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
00055 *> \endverbatim
00056 *
00057 *  Arguments:
00058 *  ==========
00059 *
00060 *> \param[in] WANTT
00061 *> \verbatim
00062 *>          WANTT is LOGICAL
00063 *>          = .TRUE. : the full Schur form T is required;
00064 *>          = .FALSE.: only eigenvalues are required.
00065 *> \endverbatim
00066 *>
00067 *> \param[in] WANTZ
00068 *> \verbatim
00069 *>          WANTZ is LOGICAL
00070 *>          = .TRUE. : the matrix of Schur vectors Z is required;
00071 *>          = .FALSE.: Schur vectors are not required.
00072 *> \endverbatim
00073 *>
00074 *> \param[in] N
00075 *> \verbatim
00076 *>          N is INTEGER
00077 *>           The order of the matrix H.  N .GE. 0.
00078 *> \endverbatim
00079 *>
00080 *> \param[in] ILO
00081 *> \verbatim
00082 *>          ILO is INTEGER
00083 *> \endverbatim
00084 *>
00085 *> \param[in] IHI
00086 *> \verbatim
00087 *>          IHI is INTEGER
00088 *>           It is assumed that H is already upper triangular in rows
00089 *>           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
00090 *>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
00091 *>           previous call to CGEBAL, and then passed to CGEHRD when the
00092 *>           matrix output by CGEBAL is reduced to Hessenberg form.
00093 *>           Otherwise, ILO and IHI should be set to 1 and N,
00094 *>           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
00095 *>           If N = 0, then ILO = 1 and IHI = 0.
00096 *> \endverbatim
00097 *>
00098 *> \param[in,out] H
00099 *> \verbatim
00100 *>          H is COMPLEX array, dimension (LDH,N)
00101 *>           On entry, the upper Hessenberg matrix H.
00102 *>           On exit, if INFO = 0 and WANTT is .TRUE., then H
00103 *>           contains the upper triangular matrix T from the Schur
00104 *>           decomposition (the Schur form). If INFO = 0 and WANT is
00105 *>           .FALSE., then the contents of H are unspecified on exit.
00106 *>           (The output value of H when INFO.GT.0 is given under the
00107 *>           description of INFO below.)
00108 *>
00109 *>           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
00110 *>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
00111 *> \endverbatim
00112 *>
00113 *> \param[in] LDH
00114 *> \verbatim
00115 *>          LDH is INTEGER
00116 *>           The leading dimension of the array H. LDH .GE. max(1,N).
00117 *> \endverbatim
00118 *>
00119 *> \param[out] W
00120 *> \verbatim
00121 *>          W is COMPLEX array, dimension (N)
00122 *>           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
00123 *>           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
00124 *>           stored in the same order as on the diagonal of the Schur
00125 *>           form returned in H, with W(i) = H(i,i).
00126 *> \endverbatim
00127 *>
00128 *> \param[in] ILOZ
00129 *> \verbatim
00130 *>          ILOZ is INTEGER
00131 *> \endverbatim
00132 *>
00133 *> \param[in] IHIZ
00134 *> \verbatim
00135 *>          IHIZ is INTEGER
00136 *>           Specify the rows of Z to which transformations must be
00137 *>           applied if WANTZ is .TRUE..
00138 *>           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
00139 *> \endverbatim
00140 *>
00141 *> \param[in,out] Z
00142 *> \verbatim
00143 *>          Z is COMPLEX array, dimension (LDZ,IHI)
00144 *>           If WANTZ is .FALSE., then Z is not referenced.
00145 *>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
00146 *>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
00147 *>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
00148 *>           (The output value of Z when INFO.GT.0 is given under
00149 *>           the description of INFO below.)
00150 *> \endverbatim
00151 *>
00152 *> \param[in] LDZ
00153 *> \verbatim
00154 *>          LDZ is INTEGER
00155 *>           The leading dimension of the array Z.  if WANTZ is .TRUE.
00156 *>           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
00157 *> \endverbatim
00158 *>
00159 *> \param[out] WORK
00160 *> \verbatim
00161 *>          WORK is COMPLEX array, dimension LWORK
00162 *>           On exit, if LWORK = -1, WORK(1) returns an estimate of
00163 *>           the optimal value for LWORK.
00164 *> \endverbatim
00165 *>
00166 *> \param[in] LWORK
00167 *> \verbatim
00168 *>          LWORK is INTEGER
00169 *>           The dimension of the array WORK.  LWORK .GE. max(1,N)
00170 *>           is sufficient, but LWORK typically as large as 6*N may
00171 *>           be required for optimal performance.  A workspace query
00172 *>           to determine the optimal workspace size is recommended.
00173 *>
00174 *>           If LWORK = -1, then CLAQR4 does a workspace query.
00175 *>           In this case, CLAQR4 checks the input parameters and
00176 *>           estimates the optimal workspace size for the given
00177 *>           values of N, ILO and IHI.  The estimate is returned
00178 *>           in WORK(1).  No error message related to LWORK is
00179 *>           issued by XERBLA.  Neither H nor Z are accessed.
00180 *> \endverbatim
00181 *>
00182 *> \param[out] INFO
00183 *> \verbatim
00184 *> \verbatim
00185 *>          INFO is INTEGER
00186 *>             =  0:  successful exit
00187 *>           .GT. 0:  if INFO = i, CLAQR4 failed to compute all of
00188 *>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00189 *>                and WI contain those eigenvalues which have been
00190 *>                successfully computed.  (Failures are rare.)
00191 *>
00192 *>                If INFO .GT. 0 and WANT is .FALSE., then on exit,
00193 *>                the remaining unconverged eigenvalues are the eigen-
00194 *>                values of the upper Hessenberg matrix rows and
00195 *>                columns ILO through INFO of the final, output
00196 *>                value of H.
00197 *>
00198 *>                If INFO .GT. 0 and WANTT is .TRUE., then on exit
00199 *>
00200 *>           (*)  (initial value of H)*U  = U*(final value of H)
00201 *>
00202 *>                where U is a unitary matrix.  The final
00203 *>                value of  H is upper Hessenberg and triangular in
00204 *>                rows and columns INFO+1 through IHI.
00205 *>
00206 *>                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
00207 *>
00208 *>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
00209 *>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
00210 *>
00211 *>                where U is the unitary matrix in (*) (regard-
00212 *>                less of the value of WANTT.)
00213 *>
00214 *>                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
00215 *>                accessed.
00216 *> \endverbatim
00217 *
00218 *  Authors:
00219 *  ========
00220 *
00221 *> \author Univ. of Tennessee 
00222 *> \author Univ. of California Berkeley 
00223 *> \author Univ. of Colorado Denver 
00224 *> \author NAG Ltd. 
00225 *
00226 *> \date November 2011
00227 *
00228 *> \ingroup complexOTHERauxiliary
00229 *
00230 *> \par Contributors:
00231 *  ==================
00232 *>
00233 *>       Karen Braman and Ralph Byers, Department of Mathematics,
00234 *>       University of Kansas, USA
00235 *
00236 *> \par References:
00237 *  ================
00238 *>
00239 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00240 *>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00241 *>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00242 *>       929--947, 2002.
00243 *> \n
00244 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00245 *>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00246 *>       of Matrix Analysis, volume 23, pages 948--973, 2002.
00247 *>
00248 *  =====================================================================
00249       SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
00250      $                   IHIZ, Z, LDZ, WORK, LWORK, INFO )
00251 *
00252 *  -- LAPACK auxiliary routine (version 3.4.0) --
00253 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00254 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00255 *     November 2011
00256 *
00257 *     .. Scalar Arguments ..
00258       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
00259       LOGICAL            WANTT, WANTZ
00260 *     ..
00261 *     .. Array Arguments ..
00262       COMPLEX            H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
00263 *     ..
00264 *
00265 *
00266 *  ================================================================
00267 *
00268 *     .. Parameters ..
00269 *
00270 *     ==== Matrices of order NTINY or smaller must be processed by
00271 *     .    CLAHQR because of insufficient subdiagonal scratch space.
00272 *     .    (This is a hard limit.) ====
00273       INTEGER            NTINY
00274       PARAMETER          ( NTINY = 11 )
00275 *
00276 *     ==== Exceptional deflation windows:  try to cure rare
00277 *     .    slow convergence by varying the size of the
00278 *     .    deflation window after KEXNW iterations. ====
00279       INTEGER            KEXNW
00280       PARAMETER          ( KEXNW = 5 )
00281 *
00282 *     ==== Exceptional shifts: try to cure rare slow convergence
00283 *     .    with ad-hoc exceptional shifts every KEXSH iterations.
00284 *     .    ====
00285       INTEGER            KEXSH
00286       PARAMETER          ( KEXSH = 6 )
00287 *
00288 *     ==== The constant WILK1 is used to form the exceptional
00289 *     .    shifts. ====
00290       REAL               WILK1
00291       PARAMETER          ( WILK1 = 0.75e0 )
00292       COMPLEX            ZERO, ONE
00293       PARAMETER          ( ZERO = ( 0.0e0, 0.0e0 ),
00294      $                   ONE = ( 1.0e0, 0.0e0 ) )
00295       REAL               TWO
00296       PARAMETER          ( TWO = 2.0e0 )
00297 *     ..
00298 *     .. Local Scalars ..
00299       COMPLEX            AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
00300       REAL               S
00301       INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
00302      $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
00303      $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
00304      $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
00305       LOGICAL            SORTED
00306       CHARACTER          JBCMPZ*2
00307 *     ..
00308 *     .. External Functions ..
00309       INTEGER            ILAENV
00310       EXTERNAL           ILAENV
00311 *     ..
00312 *     .. Local Arrays ..
00313       COMPLEX            ZDUM( 1, 1 )
00314 *     ..
00315 *     .. External Subroutines ..
00316       EXTERNAL           CLACPY, CLAHQR, CLAQR2, CLAQR5
00317 *     ..
00318 *     .. Intrinsic Functions ..
00319       INTRINSIC          ABS, AIMAG, CMPLX, INT, MAX, MIN, MOD, REAL,
00320      $                   SQRT
00321 *     ..
00322 *     .. Statement Functions ..
00323       REAL               CABS1
00324 *     ..
00325 *     .. Statement Function definitions ..
00326       CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
00327 *     ..
00328 *     .. Executable Statements ..
00329       INFO = 0
00330 *
00331 *     ==== Quick return for N = 0: nothing to do. ====
00332 *
00333       IF( N.EQ.0 ) THEN
00334          WORK( 1 ) = ONE
00335          RETURN
00336       END IF
00337 *
00338       IF( N.LE.NTINY ) THEN
00339 *
00340 *        ==== Tiny matrices must use CLAHQR. ====
00341 *
00342          LWKOPT = 1
00343          IF( LWORK.NE.-1 )
00344      $      CALL CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
00345      $                   IHIZ, Z, LDZ, INFO )
00346       ELSE
00347 *
00348 *        ==== Use small bulge multi-shift QR with aggressive early
00349 *        .    deflation on larger-than-tiny matrices. ====
00350 *
00351 *        ==== Hope for the best. ====
00352 *
00353          INFO = 0
00354 *
00355 *        ==== Set up job flags for ILAENV. ====
00356 *
00357          IF( WANTT ) THEN
00358             JBCMPZ( 1: 1 ) = 'S'
00359          ELSE
00360             JBCMPZ( 1: 1 ) = 'E'
00361          END IF
00362          IF( WANTZ ) THEN
00363             JBCMPZ( 2: 2 ) = 'V'
00364          ELSE
00365             JBCMPZ( 2: 2 ) = 'N'
00366          END IF
00367 *
00368 *        ==== NWR = recommended deflation window size.  At this
00369 *        .    point,  N .GT. NTINY = 11, so there is enough
00370 *        .    subdiagonal workspace for NWR.GE.2 as required.
00371 *        .    (In fact, there is enough subdiagonal space for
00372 *        .    NWR.GE.3.) ====
00373 *
00374          NWR = ILAENV( 13, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00375          NWR = MAX( 2, NWR )
00376          NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
00377 *
00378 *        ==== NSR = recommended number of simultaneous shifts.
00379 *        .    At this point N .GT. NTINY = 11, so there is at
00380 *        .    enough subdiagonal workspace for NSR to be even
00381 *        .    and greater than or equal to two as required. ====
00382 *
00383          NSR = ILAENV( 15, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00384          NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
00385          NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
00386 *
00387 *        ==== Estimate optimal workspace ====
00388 *
00389 *        ==== Workspace query call to CLAQR2 ====
00390 *
00391          CALL CLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
00392      $                IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H,
00393      $                LDH, WORK, -1 )
00394 *
00395 *        ==== Optimal workspace = MAX(CLAQR5, CLAQR2) ====
00396 *
00397          LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
00398 *
00399 *        ==== Quick return in case of workspace query. ====
00400 *
00401          IF( LWORK.EQ.-1 ) THEN
00402             WORK( 1 ) = CMPLX( LWKOPT, 0 )
00403             RETURN
00404          END IF
00405 *
00406 *        ==== CLAHQR/CLAQR0 crossover point ====
00407 *
00408          NMIN = ILAENV( 12, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00409          NMIN = MAX( NTINY, NMIN )
00410 *
00411 *        ==== Nibble crossover point ====
00412 *
00413          NIBBLE = ILAENV( 14, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00414          NIBBLE = MAX( 0, NIBBLE )
00415 *
00416 *        ==== Accumulate reflections during ttswp?  Use block
00417 *        .    2-by-2 structure during matrix-matrix multiply? ====
00418 *
00419          KACC22 = ILAENV( 16, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00420          KACC22 = MAX( 0, KACC22 )
00421          KACC22 = MIN( 2, KACC22 )
00422 *
00423 *        ==== NWMAX = the largest possible deflation window for
00424 *        .    which there is sufficient workspace. ====
00425 *
00426          NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
00427          NW = NWMAX
00428 *
00429 *        ==== NSMAX = the Largest number of simultaneous shifts
00430 *        .    for which there is sufficient workspace. ====
00431 *
00432          NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
00433          NSMAX = NSMAX - MOD( NSMAX, 2 )
00434 *
00435 *        ==== NDFL: an iteration count restarted at deflation. ====
00436 *
00437          NDFL = 1
00438 *
00439 *        ==== ITMAX = iteration limit ====
00440 *
00441          ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
00442 *
00443 *        ==== Last row and column in the active block ====
00444 *
00445          KBOT = IHI
00446 *
00447 *        ==== Main Loop ====
00448 *
00449          DO 70 IT = 1, ITMAX
00450 *
00451 *           ==== Done when KBOT falls below ILO ====
00452 *
00453             IF( KBOT.LT.ILO )
00454      $         GO TO 80
00455 *
00456 *           ==== Locate active block ====
00457 *
00458             DO 10 K = KBOT, ILO + 1, -1
00459                IF( H( K, K-1 ).EQ.ZERO )
00460      $            GO TO 20
00461    10       CONTINUE
00462             K = ILO
00463    20       CONTINUE
00464             KTOP = K
00465 *
00466 *           ==== Select deflation window size:
00467 *           .    Typical Case:
00468 *           .      If possible and advisable, nibble the entire
00469 *           .      active block.  If not, use size MIN(NWR,NWMAX)
00470 *           .      or MIN(NWR+1,NWMAX) depending upon which has
00471 *           .      the smaller corresponding subdiagonal entry
00472 *           .      (a heuristic).
00473 *           .
00474 *           .    Exceptional Case:
00475 *           .      If there have been no deflations in KEXNW or
00476 *           .      more iterations, then vary the deflation window
00477 *           .      size.   At first, because, larger windows are,
00478 *           .      in general, more powerful than smaller ones,
00479 *           .      rapidly increase the window to the maximum possible.
00480 *           .      Then, gradually reduce the window size. ====
00481 *
00482             NH = KBOT - KTOP + 1
00483             NWUPBD = MIN( NH, NWMAX )
00484             IF( NDFL.LT.KEXNW ) THEN
00485                NW = MIN( NWUPBD, NWR )
00486             ELSE
00487                NW = MIN( NWUPBD, 2*NW )
00488             END IF
00489             IF( NW.LT.NWMAX ) THEN
00490                IF( NW.GE.NH-1 ) THEN
00491                   NW = NH
00492                ELSE
00493                   KWTOP = KBOT - NW + 1
00494                   IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT.
00495      $                CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
00496                END IF
00497             END IF
00498             IF( NDFL.LT.KEXNW ) THEN
00499                NDEC = -1
00500             ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
00501                NDEC = NDEC + 1
00502                IF( NW-NDEC.LT.2 )
00503      $            NDEC = 0
00504                NW = NW - NDEC
00505             END IF
00506 *
00507 *           ==== Aggressive early deflation:
00508 *           .    split workspace under the subdiagonal into
00509 *           .      - an nw-by-nw work array V in the lower
00510 *           .        left-hand-corner,
00511 *           .      - an NW-by-at-least-NW-but-more-is-better
00512 *           .        (NW-by-NHO) horizontal work array along
00513 *           .        the bottom edge,
00514 *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
00515 *           .        vertical work array along the left-hand-edge.
00516 *           .        ====
00517 *
00518             KV = N - NW + 1
00519             KT = NW + 1
00520             NHO = ( N-NW-1 ) - KT + 1
00521             KWV = NW + 2
00522             NVE = ( N-NW ) - KWV + 1
00523 *
00524 *           ==== Aggressive early deflation ====
00525 *
00526             CALL CLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
00527      $                   IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO,
00528      $                   H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK,
00529      $                   LWORK )
00530 *
00531 *           ==== Adjust KBOT accounting for new deflations. ====
00532 *
00533             KBOT = KBOT - LD
00534 *
00535 *           ==== KS points to the shifts. ====
00536 *
00537             KS = KBOT - LS + 1
00538 *
00539 *           ==== Skip an expensive QR sweep if there is a (partly
00540 *           .    heuristic) reason to expect that many eigenvalues
00541 *           .    will deflate without it.  Here, the QR sweep is
00542 *           .    skipped if many eigenvalues have just been deflated
00543 *           .    or if the remaining active block is small.
00544 *
00545             IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
00546      $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
00547 *
00548 *              ==== NS = nominal number of simultaneous shifts.
00549 *              .    This may be lowered (slightly) if CLAQR2
00550 *              .    did not provide that many shifts. ====
00551 *
00552                NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
00553                NS = NS - MOD( NS, 2 )
00554 *
00555 *              ==== If there have been no deflations
00556 *              .    in a multiple of KEXSH iterations,
00557 *              .    then try exceptional shifts.
00558 *              .    Otherwise use shifts provided by
00559 *              .    CLAQR2 above or from the eigenvalues
00560 *              .    of a trailing principal submatrix. ====
00561 *
00562                IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
00563                   KS = KBOT - NS + 1
00564                   DO 30 I = KBOT, KS + 1, -2
00565                      W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) )
00566                      W( I-1 ) = W( I )
00567    30             CONTINUE
00568                ELSE
00569 *
00570 *                 ==== Got NS/2 or fewer shifts? Use CLAHQR
00571 *                 .    on a trailing principal submatrix to
00572 *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
00573 *                 .    there is enough space below the subdiagonal
00574 *                 .    to fit an NS-by-NS scratch array.) ====
00575 *
00576                   IF( KBOT-KS+1.LE.NS / 2 ) THEN
00577                      KS = KBOT - NS + 1
00578                      KT = N - NS + 1
00579                      CALL CLACPY( 'A', NS, NS, H( KS, KS ), LDH,
00580      $                            H( KT, 1 ), LDH )
00581                      CALL CLAHQR( .false., .false., NS, 1, NS,
00582      $                            H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM,
00583      $                            1, INF )
00584                      KS = KS + INF
00585 *
00586 *                    ==== In case of a rare QR failure use
00587 *                    .    eigenvalues of the trailing 2-by-2
00588 *                    .    principal submatrix.  Scale to avoid
00589 *                    .    overflows, underflows and subnormals.
00590 *                    .    (The scale factor S can not be zero,
00591 *                    .    because H(KBOT,KBOT-1) is nonzero.) ====
00592 *
00593                      IF( KS.GE.KBOT ) THEN
00594                         S = CABS1( H( KBOT-1, KBOT-1 ) ) +
00595      $                      CABS1( H( KBOT, KBOT-1 ) ) +
00596      $                      CABS1( H( KBOT-1, KBOT ) ) +
00597      $                      CABS1( H( KBOT, KBOT ) )
00598                         AA = H( KBOT-1, KBOT-1 ) / S
00599                         CC = H( KBOT, KBOT-1 ) / S
00600                         BB = H( KBOT-1, KBOT ) / S
00601                         DD = H( KBOT, KBOT ) / S
00602                         TR2 = ( AA+DD ) / TWO
00603                         DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC
00604                         RTDISC = SQRT( -DET )
00605                         W( KBOT-1 ) = ( TR2+RTDISC )*S
00606                         W( KBOT ) = ( TR2-RTDISC )*S
00607 *
00608                         KS = KBOT - 1
00609                      END IF
00610                   END IF
00611 *
00612                   IF( KBOT-KS+1.GT.NS ) THEN
00613 *
00614 *                    ==== Sort the shifts (Helps a little) ====
00615 *
00616                      SORTED = .false.
00617                      DO 50 K = KBOT, KS + 1, -1
00618                         IF( SORTED )
00619      $                     GO TO 60
00620                         SORTED = .true.
00621                         DO 40 I = KS, K - 1
00622                            IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) )
00623      $                          THEN
00624                               SORTED = .false.
00625                               SWAP = W( I )
00626                               W( I ) = W( I+1 )
00627                               W( I+1 ) = SWAP
00628                            END IF
00629    40                   CONTINUE
00630    50                CONTINUE
00631    60                CONTINUE
00632                   END IF
00633                END IF
00634 *
00635 *              ==== If there are only two shifts, then use
00636 *              .    only one.  ====
00637 *
00638                IF( KBOT-KS+1.EQ.2 ) THEN
00639                   IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT.
00640      $                CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
00641                      W( KBOT-1 ) = W( KBOT )
00642                   ELSE
00643                      W( KBOT ) = W( KBOT-1 )
00644                   END IF
00645                END IF
00646 *
00647 *              ==== Use up to NS of the the smallest magnatiude
00648 *              .    shifts.  If there aren't NS shifts available,
00649 *              .    then use them all, possibly dropping one to
00650 *              .    make the number of shifts even. ====
00651 *
00652                NS = MIN( NS, KBOT-KS+1 )
00653                NS = NS - MOD( NS, 2 )
00654                KS = KBOT - NS + 1
00655 *
00656 *              ==== Small-bulge multi-shift QR sweep:
00657 *              .    split workspace under the subdiagonal into
00658 *              .    - a KDU-by-KDU work array U in the lower
00659 *              .      left-hand-corner,
00660 *              .    - a KDU-by-at-least-KDU-but-more-is-better
00661 *              .      (KDU-by-NHo) horizontal work array WH along
00662 *              .      the bottom edge,
00663 *              .    - and an at-least-KDU-but-more-is-better-by-KDU
00664 *              .      (NVE-by-KDU) vertical work WV arrow along
00665 *              .      the left-hand-edge. ====
00666 *
00667                KDU = 3*NS - 3
00668                KU = N - KDU + 1
00669                KWH = KDU + 1
00670                NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
00671                KWV = KDU + 4
00672                NVE = N - KDU - KWV + 1
00673 *
00674 *              ==== Small-bulge multi-shift QR sweep ====
00675 *
00676                CALL CLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
00677      $                      W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK,
00678      $                      3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH,
00679      $                      NHO, H( KU, KWH ), LDH )
00680             END IF
00681 *
00682 *           ==== Note progress (or the lack of it). ====
00683 *
00684             IF( LD.GT.0 ) THEN
00685                NDFL = 1
00686             ELSE
00687                NDFL = NDFL + 1
00688             END IF
00689 *
00690 *           ==== End of main loop ====
00691    70    CONTINUE
00692 *
00693 *        ==== Iteration limit exceeded.  Set INFO to show where
00694 *        .    the problem occurred and exit. ====
00695 *
00696          INFO = KBOT
00697    80    CONTINUE
00698       END IF
00699 *
00700 *     ==== Return the optimal value of LWORK. ====
00701 *
00702       WORK( 1 ) = CMPLX( LWKOPT, 0 )
00703 *
00704 *     ==== End of CLAQR4 ====
00705 *
00706       END
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