LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
slaqr0.f
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00001 *> \brief \b SLAQR0
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download SLAQR0 + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00022 *                          ILOZ, 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 *       REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
00030 *      $                   Z( LDZ, * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *>    SLAQR0 computes the eigenvalues of a Hessenberg matrix H
00040 *>    and, optionally, the matrices T and Z from the Schur decomposition
00041 *>    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
00042 *>    Schur form), and Z is the orthogonal matrix of Schur vectors.
00043 *>
00044 *>    Optionally Z may be postmultiplied into an input orthogonal
00045 *>    matrix Q so that this routine can give the Schur factorization
00046 *>    of a matrix A which has been reduced to the Hessenberg form H
00047 *>    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
00048 *> \endverbatim
00049 *
00050 *  Arguments:
00051 *  ==========
00052 *
00053 *> \param[in] WANTT
00054 *> \verbatim
00055 *>          WANTT is LOGICAL
00056 *>          = .TRUE. : the full Schur form T is required;
00057 *>          = .FALSE.: only eigenvalues are required.
00058 *> \endverbatim
00059 *>
00060 *> \param[in] WANTZ
00061 *> \verbatim
00062 *>          WANTZ is LOGICAL
00063 *>          = .TRUE. : the matrix of Schur vectors Z is required;
00064 *>          = .FALSE.: Schur vectors are not required.
00065 *> \endverbatim
00066 *>
00067 *> \param[in] N
00068 *> \verbatim
00069 *>          N is INTEGER
00070 *>           The order of the matrix H.  N .GE. 0.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] ILO
00074 *> \verbatim
00075 *>          ILO is INTEGER
00076 *> \endverbatim
00077 *>
00078 *> \param[in] IHI
00079 *> \verbatim
00080 *>          IHI is INTEGER
00081 *>           It is assumed that H is already upper triangular in rows
00082 *>           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
00083 *>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
00084 *>           previous call to SGEBAL, and then passed to SGEHRD when the
00085 *>           matrix output by SGEBAL is reduced to Hessenberg form.
00086 *>           Otherwise, ILO and IHI should be set to 1 and N,
00087 *>           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
00088 *>           If N = 0, then ILO = 1 and IHI = 0.
00089 *> \endverbatim
00090 *>
00091 *> \param[in,out] H
00092 *> \verbatim
00093 *>          H is REAL array, dimension (LDH,N)
00094 *>           On entry, the upper Hessenberg matrix H.
00095 *>           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
00096 *>           the upper quasi-triangular matrix T from the Schur
00097 *>           decomposition (the Schur form); 2-by-2 diagonal blocks
00098 *>           (corresponding to complex conjugate pairs of eigenvalues)
00099 *>           are returned in standard form, with H(i,i) = H(i+1,i+1)
00100 *>           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
00101 *>           .FALSE., then the contents of H are unspecified on exit.
00102 *>           (The output value of H when INFO.GT.0 is given under the
00103 *>           description of INFO below.)
00104 *>
00105 *>           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
00106 *>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
00107 *> \endverbatim
00108 *>
00109 *> \param[in] LDH
00110 *> \verbatim
00111 *>          LDH is INTEGER
00112 *>           The leading dimension of the array H. LDH .GE. max(1,N).
00113 *> \endverbatim
00114 *>
00115 *> \param[out] WR
00116 *> \verbatim
00117 *>          WR is REAL array, dimension (IHI)
00118 *> \endverbatim
00119 *>
00120 *> \param[out] WI
00121 *> \verbatim
00122 *>          WI is REAL array, dimension (IHI)
00123 *>           The real and imaginary parts, respectively, of the computed
00124 *>           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
00125 *>           and WI(ILO:IHI). If two eigenvalues are computed as a
00126 *>           complex conjugate pair, they are stored in consecutive
00127 *>           elements of WR and WI, say the i-th and (i+1)th, with
00128 *>           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
00129 *>           the eigenvalues are stored in the same order as on the
00130 *>           diagonal of the Schur form returned in H, with
00131 *>           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
00132 *>           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
00133 *>           WI(i+1) = -WI(i).
00134 *> \endverbatim
00135 *>
00136 *> \param[in] ILOZ
00137 *> \verbatim
00138 *>          ILOZ is INTEGER
00139 *> \endverbatim
00140 *>
00141 *> \param[in] IHIZ
00142 *> \verbatim
00143 *>          IHIZ is INTEGER
00144 *>           Specify the rows of Z to which transformations must be
00145 *>           applied if WANTZ is .TRUE..
00146 *>           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
00147 *> \endverbatim
00148 *>
00149 *> \param[in,out] Z
00150 *> \verbatim
00151 *>          Z is REAL array, dimension (LDZ,IHI)
00152 *>           If WANTZ is .FALSE., then Z is not referenced.
00153 *>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
00154 *>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
00155 *>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
00156 *>           (The output value of Z when INFO.GT.0 is given under
00157 *>           the description of INFO below.)
00158 *> \endverbatim
00159 *>
00160 *> \param[in] LDZ
00161 *> \verbatim
00162 *>          LDZ is INTEGER
00163 *>           The leading dimension of the array Z.  if WANTZ is .TRUE.
00164 *>           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
00165 *> \endverbatim
00166 *>
00167 *> \param[out] WORK
00168 *> \verbatim
00169 *>          WORK is REAL array, dimension LWORK
00170 *>           On exit, if LWORK = -1, WORK(1) returns an estimate of
00171 *>           the optimal value for LWORK.
00172 *> \endverbatim
00173 *>
00174 *> \param[in] LWORK
00175 *> \verbatim
00176 *>          LWORK is INTEGER
00177 *>           The dimension of the array WORK.  LWORK .GE. max(1,N)
00178 *>           is sufficient, but LWORK typically as large as 6*N may
00179 *>           be required for optimal performance.  A workspace query
00180 *>           to determine the optimal workspace size is recommended.
00181 *>
00182 *>           If LWORK = -1, then SLAQR0 does a workspace query.
00183 *>           In this case, SLAQR0 checks the input parameters and
00184 *>           estimates the optimal workspace size for the given
00185 *>           values of N, ILO and IHI.  The estimate is returned
00186 *>           in WORK(1).  No error message related to LWORK is
00187 *>           issued by XERBLA.  Neither H nor Z are accessed.
00188 *> \endverbatim
00189 *>
00190 *> \param[out] INFO
00191 *> \verbatim
00192 *>          INFO is INTEGER
00193 *>             =  0:  successful exit
00194 *>           .GT. 0:  if INFO = i, SLAQR0 failed to compute all of
00195 *>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00196 *>                and WI contain those eigenvalues which have been
00197 *>                successfully computed.  (Failures are rare.)
00198 *>
00199 *>                If INFO .GT. 0 and WANT is .FALSE., then on exit,
00200 *>                the remaining unconverged eigenvalues are the eigen-
00201 *>                values of the upper Hessenberg matrix rows and
00202 *>                columns ILO through INFO of the final, output
00203 *>                value of H.
00204 *>
00205 *>                If INFO .GT. 0 and WANTT is .TRUE., then on exit
00206 *>
00207 *>           (*)  (initial value of H)*U  = U*(final value of H)
00208 *>
00209 *>                where U is an orthogonal matrix.  The final
00210 *>                value of H is upper Hessenberg and quasi-triangular
00211 *>                in rows and columns INFO+1 through IHI.
00212 *>
00213 *>                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
00214 *>
00215 *>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
00216 *>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
00217 *>
00218 *>                where U is the orthogonal matrix in (*) (regard-
00219 *>                less of the value of WANTT.)
00220 *>
00221 *>                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
00222 *>                accessed.
00223 *> \endverbatim
00224 *
00225 *  Authors:
00226 *  ========
00227 *
00228 *> \author Univ. of Tennessee 
00229 *> \author Univ. of California Berkeley 
00230 *> \author Univ. of Colorado Denver 
00231 *> \author NAG Ltd. 
00232 *
00233 *> \date November 2011
00234 *
00235 *> \ingroup realOTHERauxiliary
00236 *
00237 *> \par Contributors:
00238 *  ==================
00239 *>
00240 *>       Karen Braman and Ralph Byers, Department of Mathematics,
00241 *>       University of Kansas, USA
00242 *
00243 *> \par References:
00244 *  ================
00245 *>
00246 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00247 *>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00248 *>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00249 *>       929--947, 2002.
00250 *> \n
00251 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00252 *>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00253 *>       of Matrix Analysis, volume 23, pages 948--973, 2002.
00254 *>
00255 *  =====================================================================
00256       SUBROUTINE SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00257      $                   ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
00258 *
00259 *  -- LAPACK auxiliary routine (version 3.4.0) --
00260 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00261 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00262 *     November 2011
00263 *
00264 *     .. Scalar Arguments ..
00265       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
00266       LOGICAL            WANTT, WANTZ
00267 *     ..
00268 *     .. Array Arguments ..
00269       REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
00270      $                   Z( LDZ, * )
00271 *     ..
00272 *
00273 *  ================================================================
00274 *     .. Parameters ..
00275 *
00276 *     ==== Matrices of order NTINY or smaller must be processed by
00277 *     .    SLAHQR because of insufficient subdiagonal scratch space.
00278 *     .    (This is a hard limit.) ====
00279       INTEGER            NTINY
00280       PARAMETER          ( NTINY = 11 )
00281 *
00282 *     ==== Exceptional deflation windows:  try to cure rare
00283 *     .    slow convergence by varying the size of the
00284 *     .    deflation window after KEXNW iterations. ====
00285       INTEGER            KEXNW
00286       PARAMETER          ( KEXNW = 5 )
00287 *
00288 *     ==== Exceptional shifts: try to cure rare slow convergence
00289 *     .    with ad-hoc exceptional shifts every KEXSH iterations.
00290 *     .    ====
00291       INTEGER            KEXSH
00292       PARAMETER          ( KEXSH = 6 )
00293 *
00294 *     ==== The constants WILK1 and WILK2 are used to form the
00295 *     .    exceptional shifts. ====
00296       REAL               WILK1, WILK2
00297       PARAMETER          ( WILK1 = 0.75e0, WILK2 = -0.4375e0 )
00298       REAL               ZERO, ONE
00299       PARAMETER          ( ZERO = 0.0e0, ONE = 1.0e0 )
00300 *     ..
00301 *     .. Local Scalars ..
00302       REAL               AA, BB, CC, CS, DD, SN, SS, SWAP
00303       INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
00304      $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
00305      $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
00306      $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
00307       LOGICAL            SORTED
00308       CHARACTER          JBCMPZ*2
00309 *     ..
00310 *     .. External Functions ..
00311       INTEGER            ILAENV
00312       EXTERNAL           ILAENV
00313 *     ..
00314 *     .. Local Arrays ..
00315       REAL               ZDUM( 1, 1 )
00316 *     ..
00317 *     .. External Subroutines ..
00318       EXTERNAL           SLACPY, SLAHQR, SLANV2, SLAQR3, SLAQR4, SLAQR5
00319 *     ..
00320 *     .. Intrinsic Functions ..
00321       INTRINSIC          ABS, INT, MAX, MIN, MOD, REAL
00322 *     ..
00323 *     .. Executable Statements ..
00324       INFO = 0
00325 *
00326 *     ==== Quick return for N = 0: nothing to do. ====
00327 *
00328       IF( N.EQ.0 ) THEN
00329          WORK( 1 ) = ONE
00330          RETURN
00331       END IF
00332 *
00333       IF( N.LE.NTINY ) THEN
00334 *
00335 *        ==== Tiny matrices must use SLAHQR. ====
00336 *
00337          LWKOPT = 1
00338          IF( LWORK.NE.-1 )
00339      $      CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00340      $                   ILOZ, IHIZ, Z, LDZ, INFO )
00341       ELSE
00342 *
00343 *        ==== Use small bulge multi-shift QR with aggressive early
00344 *        .    deflation on larger-than-tiny matrices. ====
00345 *
00346 *        ==== Hope for the best. ====
00347 *
00348          INFO = 0
00349 *
00350 *        ==== Set up job flags for ILAENV. ====
00351 *
00352          IF( WANTT ) THEN
00353             JBCMPZ( 1: 1 ) = 'S'
00354          ELSE
00355             JBCMPZ( 1: 1 ) = 'E'
00356          END IF
00357          IF( WANTZ ) THEN
00358             JBCMPZ( 2: 2 ) = 'V'
00359          ELSE
00360             JBCMPZ( 2: 2 ) = 'N'
00361          END IF
00362 *
00363 *        ==== NWR = recommended deflation window size.  At this
00364 *        .    point,  N .GT. NTINY = 11, so there is enough
00365 *        .    subdiagonal workspace for NWR.GE.2 as required.
00366 *        .    (In fact, there is enough subdiagonal space for
00367 *        .    NWR.GE.3.) ====
00368 *
00369          NWR = ILAENV( 13, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
00370          NWR = MAX( 2, NWR )
00371          NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
00372 *
00373 *        ==== NSR = recommended number of simultaneous shifts.
00374 *        .    At this point N .GT. NTINY = 11, so there is at
00375 *        .    enough subdiagonal workspace for NSR to be even
00376 *        .    and greater than or equal to two as required. ====
00377 *
00378          NSR = ILAENV( 15, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
00379          NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
00380          NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
00381 *
00382 *        ==== Estimate optimal workspace ====
00383 *
00384 *        ==== Workspace query call to SLAQR3 ====
00385 *
00386          CALL SLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
00387      $                IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
00388      $                N, H, LDH, WORK, -1 )
00389 *
00390 *        ==== Optimal workspace = MAX(SLAQR5, SLAQR3) ====
00391 *
00392          LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
00393 *
00394 *        ==== Quick return in case of workspace query. ====
00395 *
00396          IF( LWORK.EQ.-1 ) THEN
00397             WORK( 1 ) = REAL( LWKOPT )
00398             RETURN
00399          END IF
00400 *
00401 *        ==== SLAHQR/SLAQR0 crossover point ====
00402 *
00403          NMIN = ILAENV( 12, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
00404          NMIN = MAX( NTINY, NMIN )
00405 *
00406 *        ==== Nibble crossover point ====
00407 *
00408          NIBBLE = ILAENV( 14, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
00409          NIBBLE = MAX( 0, NIBBLE )
00410 *
00411 *        ==== Accumulate reflections during ttswp?  Use block
00412 *        .    2-by-2 structure during matrix-matrix multiply? ====
00413 *
00414          KACC22 = ILAENV( 16, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
00415          KACC22 = MAX( 0, KACC22 )
00416          KACC22 = MIN( 2, KACC22 )
00417 *
00418 *        ==== NWMAX = the largest possible deflation window for
00419 *        .    which there is sufficient workspace. ====
00420 *
00421          NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
00422          NW = NWMAX
00423 *
00424 *        ==== NSMAX = the Largest number of simultaneous shifts
00425 *        .    for which there is sufficient workspace. ====
00426 *
00427          NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
00428          NSMAX = NSMAX - MOD( NSMAX, 2 )
00429 *
00430 *        ==== NDFL: an iteration count restarted at deflation. ====
00431 *
00432          NDFL = 1
00433 *
00434 *        ==== ITMAX = iteration limit ====
00435 *
00436          ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
00437 *
00438 *        ==== Last row and column in the active block ====
00439 *
00440          KBOT = IHI
00441 *
00442 *        ==== Main Loop ====
00443 *
00444          DO 80 IT = 1, ITMAX
00445 *
00446 *           ==== Done when KBOT falls below ILO ====
00447 *
00448             IF( KBOT.LT.ILO )
00449      $         GO TO 90
00450 *
00451 *           ==== Locate active block ====
00452 *
00453             DO 10 K = KBOT, ILO + 1, -1
00454                IF( H( K, K-1 ).EQ.ZERO )
00455      $            GO TO 20
00456    10       CONTINUE
00457             K = ILO
00458    20       CONTINUE
00459             KTOP = K
00460 *
00461 *           ==== Select deflation window size:
00462 *           .    Typical Case:
00463 *           .      If possible and advisable, nibble the entire
00464 *           .      active block.  If not, use size MIN(NWR,NWMAX)
00465 *           .      or MIN(NWR+1,NWMAX) depending upon which has
00466 *           .      the smaller corresponding subdiagonal entry
00467 *           .      (a heuristic).
00468 *           .
00469 *           .    Exceptional Case:
00470 *           .      If there have been no deflations in KEXNW or
00471 *           .      more iterations, then vary the deflation window
00472 *           .      size.   At first, because, larger windows are,
00473 *           .      in general, more powerful than smaller ones,
00474 *           .      rapidly increase the window to the maximum possible.
00475 *           .      Then, gradually reduce the window size. ====
00476 *
00477             NH = KBOT - KTOP + 1
00478             NWUPBD = MIN( NH, NWMAX )
00479             IF( NDFL.LT.KEXNW ) THEN
00480                NW = MIN( NWUPBD, NWR )
00481             ELSE
00482                NW = MIN( NWUPBD, 2*NW )
00483             END IF
00484             IF( NW.LT.NWMAX ) THEN
00485                IF( NW.GE.NH-1 ) THEN
00486                   NW = NH
00487                ELSE
00488                   KWTOP = KBOT - NW + 1
00489                   IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
00490      $                ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
00491                END IF
00492             END IF
00493             IF( NDFL.LT.KEXNW ) THEN
00494                NDEC = -1
00495             ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
00496                NDEC = NDEC + 1
00497                IF( NW-NDEC.LT.2 )
00498      $            NDEC = 0
00499                NW = NW - NDEC
00500             END IF
00501 *
00502 *           ==== Aggressive early deflation:
00503 *           .    split workspace under the subdiagonal into
00504 *           .      - an nw-by-nw work array V in the lower
00505 *           .        left-hand-corner,
00506 *           .      - an NW-by-at-least-NW-but-more-is-better
00507 *           .        (NW-by-NHO) horizontal work array along
00508 *           .        the bottom edge,
00509 *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
00510 *           .        vertical work array along the left-hand-edge.
00511 *           .        ====
00512 *
00513             KV = N - NW + 1
00514             KT = NW + 1
00515             NHO = ( N-NW-1 ) - KT + 1
00516             KWV = NW + 2
00517             NVE = ( N-NW ) - KWV + 1
00518 *
00519 *           ==== Aggressive early deflation ====
00520 *
00521             CALL SLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
00522      $                   IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
00523      $                   NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
00524      $                   WORK, LWORK )
00525 *
00526 *           ==== Adjust KBOT accounting for new deflations. ====
00527 *
00528             KBOT = KBOT - LD
00529 *
00530 *           ==== KS points to the shifts. ====
00531 *
00532             KS = KBOT - LS + 1
00533 *
00534 *           ==== Skip an expensive QR sweep if there is a (partly
00535 *           .    heuristic) reason to expect that many eigenvalues
00536 *           .    will deflate without it.  Here, the QR sweep is
00537 *           .    skipped if many eigenvalues have just been deflated
00538 *           .    or if the remaining active block is small.
00539 *
00540             IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
00541      $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
00542 *
00543 *              ==== NS = nominal number of simultaneous shifts.
00544 *              .    This may be lowered (slightly) if SLAQR3
00545 *              .    did not provide that many shifts. ====
00546 *
00547                NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
00548                NS = NS - MOD( NS, 2 )
00549 *
00550 *              ==== If there have been no deflations
00551 *              .    in a multiple of KEXSH iterations,
00552 *              .    then try exceptional shifts.
00553 *              .    Otherwise use shifts provided by
00554 *              .    SLAQR3 above or from the eigenvalues
00555 *              .    of a trailing principal submatrix. ====
00556 *
00557                IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
00558                   KS = KBOT - NS + 1
00559                   DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
00560                      SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
00561                      AA = WILK1*SS + H( I, I )
00562                      BB = SS
00563                      CC = WILK2*SS
00564                      DD = AA
00565                      CALL SLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
00566      $                            WR( I ), WI( I ), CS, SN )
00567    30             CONTINUE
00568                   IF( KS.EQ.KTOP ) THEN
00569                      WR( KS+1 ) = H( KS+1, KS+1 )
00570                      WI( KS+1 ) = ZERO
00571                      WR( KS ) = WR( KS+1 )
00572                      WI( KS ) = WI( KS+1 )
00573                   END IF
00574                ELSE
00575 *
00576 *                 ==== Got NS/2 or fewer shifts? Use SLAQR4 or
00577 *                 .    SLAHQR on a trailing principal submatrix to
00578 *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
00579 *                 .    there is enough space below the subdiagonal
00580 *                 .    to fit an NS-by-NS scratch array.) ====
00581 *
00582                   IF( KBOT-KS+1.LE.NS / 2 ) THEN
00583                      KS = KBOT - NS + 1
00584                      KT = N - NS + 1
00585                      CALL SLACPY( 'A', NS, NS, H( KS, KS ), LDH,
00586      $                            H( KT, 1 ), LDH )
00587                      IF( NS.GT.NMIN ) THEN
00588                         CALL SLAQR4( .false., .false., NS, 1, NS,
00589      $                               H( KT, 1 ), LDH, WR( KS ),
00590      $                               WI( KS ), 1, 1, ZDUM, 1, WORK,
00591      $                               LWORK, INF )
00592                      ELSE
00593                         CALL SLAHQR( .false., .false., NS, 1, NS,
00594      $                               H( KT, 1 ), LDH, WR( KS ),
00595      $                               WI( KS ), 1, 1, ZDUM, 1, INF )
00596                      END IF
00597                      KS = KS + INF
00598 *
00599 *                    ==== In case of a rare QR failure use
00600 *                    .    eigenvalues of the trailing 2-by-2
00601 *                    .    principal submatrix.  ====
00602 *
00603                      IF( KS.GE.KBOT ) THEN
00604                         AA = H( KBOT-1, KBOT-1 )
00605                         CC = H( KBOT, KBOT-1 )
00606                         BB = H( KBOT-1, KBOT )
00607                         DD = H( KBOT, KBOT )
00608                         CALL SLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
00609      $                               WI( KBOT-1 ), WR( KBOT ),
00610      $                               WI( KBOT ), CS, SN )
00611                         KS = KBOT - 1
00612                      END IF
00613                   END IF
00614 *
00615                   IF( KBOT-KS+1.GT.NS ) THEN
00616 *
00617 *                    ==== Sort the shifts (Helps a little)
00618 *                    .    Bubble sort keeps complex conjugate
00619 *                    .    pairs together. ====
00620 *
00621                      SORTED = .false.
00622                      DO 50 K = KBOT, KS + 1, -1
00623                         IF( SORTED )
00624      $                     GO TO 60
00625                         SORTED = .true.
00626                         DO 40 I = KS, K - 1
00627                            IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
00628      $                         ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
00629                               SORTED = .false.
00630 *
00631                               SWAP = WR( I )
00632                               WR( I ) = WR( I+1 )
00633                               WR( I+1 ) = SWAP
00634 *
00635                               SWAP = WI( I )
00636                               WI( I ) = WI( I+1 )
00637                               WI( I+1 ) = SWAP
00638                            END IF
00639    40                   CONTINUE
00640    50                CONTINUE
00641    60                CONTINUE
00642                   END IF
00643 *
00644 *                 ==== Shuffle shifts into pairs of real shifts
00645 *                 .    and pairs of complex conjugate shifts
00646 *                 .    assuming complex conjugate shifts are
00647 *                 .    already adjacent to one another. (Yes,
00648 *                 .    they are.)  ====
00649 *
00650                   DO 70 I = KBOT, KS + 2, -2
00651                      IF( WI( I ).NE.-WI( I-1 ) ) THEN
00652 *
00653                         SWAP = WR( I )
00654                         WR( I ) = WR( I-1 )
00655                         WR( I-1 ) = WR( I-2 )
00656                         WR( I-2 ) = SWAP
00657 *
00658                         SWAP = WI( I )
00659                         WI( I ) = WI( I-1 )
00660                         WI( I-1 ) = WI( I-2 )
00661                         WI( I-2 ) = SWAP
00662                      END IF
00663    70             CONTINUE
00664                END IF
00665 *
00666 *              ==== If there are only two shifts and both are
00667 *              .    real, then use only one.  ====
00668 *
00669                IF( KBOT-KS+1.EQ.2 ) THEN
00670                   IF( WI( KBOT ).EQ.ZERO ) THEN
00671                      IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
00672      $                   ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
00673                         WR( KBOT-1 ) = WR( KBOT )
00674                      ELSE
00675                         WR( KBOT ) = WR( KBOT-1 )
00676                      END IF
00677                   END IF
00678                END IF
00679 *
00680 *              ==== Use up to NS of the the smallest magnatiude
00681 *              .    shifts.  If there aren't NS shifts available,
00682 *              .    then use them all, possibly dropping one to
00683 *              .    make the number of shifts even. ====
00684 *
00685                NS = MIN( NS, KBOT-KS+1 )
00686                NS = NS - MOD( NS, 2 )
00687                KS = KBOT - NS + 1
00688 *
00689 *              ==== Small-bulge multi-shift QR sweep:
00690 *              .    split workspace under the subdiagonal into
00691 *              .    - a KDU-by-KDU work array U in the lower
00692 *              .      left-hand-corner,
00693 *              .    - a KDU-by-at-least-KDU-but-more-is-better
00694 *              .      (KDU-by-NHo) horizontal work array WH along
00695 *              .      the bottom edge,
00696 *              .    - and an at-least-KDU-but-more-is-better-by-KDU
00697 *              .      (NVE-by-KDU) vertical work WV arrow along
00698 *              .      the left-hand-edge. ====
00699 *
00700                KDU = 3*NS - 3
00701                KU = N - KDU + 1
00702                KWH = KDU + 1
00703                NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
00704                KWV = KDU + 4
00705                NVE = N - KDU - KWV + 1
00706 *
00707 *              ==== Small-bulge multi-shift QR sweep ====
00708 *
00709                CALL SLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
00710      $                      WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
00711      $                      LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
00712      $                      H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
00713             END IF
00714 *
00715 *           ==== Note progress (or the lack of it). ====
00716 *
00717             IF( LD.GT.0 ) THEN
00718                NDFL = 1
00719             ELSE
00720                NDFL = NDFL + 1
00721             END IF
00722 *
00723 *           ==== End of main loop ====
00724    80    CONTINUE
00725 *
00726 *        ==== Iteration limit exceeded.  Set INFO to show where
00727 *        .    the problem occurred and exit. ====
00728 *
00729          INFO = KBOT
00730    90    CONTINUE
00731       END IF
00732 *
00733 *     ==== Return the optimal value of LWORK. ====
00734 *
00735       WORK( 1 ) = REAL( LWKOPT )
00736 *
00737 *     ==== End of SLAQR0 ====
00738 *
00739       END
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