LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
slaqr4.f
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00001 *> \brief \b SLAQR4
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SLAQR4 + 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/slaqr4.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SLAQR4( 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 *>    SLAQR4 implements one level of recursion for SLAQR0.
00040 *>    It is a complete implementation of the small bulge multi-shift
00041 *>    QR algorithm.  It may be called by SLAQR0 and, for large enough
00042 *>    deflation window size, it may be called by SLAQR3.  This
00043 *>    subroutine is identical to SLAQR0 except that it calls SLAQR2
00044 *>    instead of SLAQR3.
00045 *>
00046 *>    SLAQR4 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**T, where T is an upper quasi-triangular matrix (the
00049 *>    Schur form), and Z is the orthogonal matrix of Schur vectors.
00050 *>
00051 *>    Optionally Z may be postmultiplied into an input orthogonal
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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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 SGEBAL, and then passed to SGEHRD when the
00092 *>           matrix output by SGEBAL 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 REAL array, dimension (LDH,N)
00101 *>           On entry, the upper Hessenberg matrix H.
00102 *>           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
00103 *>           the upper quasi-triangular matrix T from the Schur
00104 *>           decomposition (the Schur form); 2-by-2 diagonal blocks
00105 *>           (corresponding to complex conjugate pairs of eigenvalues)
00106 *>           are returned in standard form, with H(i,i) = H(i+1,i+1)
00107 *>           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
00108 *>           .FALSE., then the contents of H are unspecified on exit.
00109 *>           (The output value of H when INFO.GT.0 is given under the
00110 *>           description of INFO below.)
00111 *>
00112 *>           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
00113 *>           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
00114 *> \endverbatim
00115 *>
00116 *> \param[in] LDH
00117 *> \verbatim
00118 *>          LDH is INTEGER
00119 *>           The leading dimension of the array H. LDH .GE. max(1,N).
00120 *> \endverbatim
00121 *>
00122 *> \param[out] WR
00123 *> \verbatim
00124 *>          WR is REAL array, dimension (IHI)
00125 *> \endverbatim
00126 *>
00127 *> \param[out] WI
00128 *> \verbatim
00129 *>          WI is REAL array, dimension (IHI)
00130 *>           The real and imaginary parts, respectively, of the computed
00131 *>           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
00132 *>           and WI(ILO:IHI). If two eigenvalues are computed as a
00133 *>           complex conjugate pair, they are stored in consecutive
00134 *>           elements of WR and WI, say the i-th and (i+1)th, with
00135 *>           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
00136 *>           the eigenvalues are stored in the same order as on the
00137 *>           diagonal of the Schur form returned in H, with
00138 *>           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
00139 *>           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
00140 *>           WI(i+1) = -WI(i).
00141 *> \endverbatim
00142 *>
00143 *> \param[in] ILOZ
00144 *> \verbatim
00145 *>          ILOZ is INTEGER
00146 *> \endverbatim
00147 *>
00148 *> \param[in] IHIZ
00149 *> \verbatim
00150 *>          IHIZ is INTEGER
00151 *>           Specify the rows of Z to which transformations must be
00152 *>           applied if WANTZ is .TRUE..
00153 *>           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
00154 *> \endverbatim
00155 *>
00156 *> \param[in,out] Z
00157 *> \verbatim
00158 *>          Z is REAL array, dimension (LDZ,IHI)
00159 *>           If WANTZ is .FALSE., then Z is not referenced.
00160 *>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
00161 *>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
00162 *>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
00163 *>           (The output value of Z when INFO.GT.0 is given under
00164 *>           the description of INFO below.)
00165 *> \endverbatim
00166 *>
00167 *> \param[in] LDZ
00168 *> \verbatim
00169 *>          LDZ is INTEGER
00170 *>           The leading dimension of the array Z.  if WANTZ is .TRUE.
00171 *>           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
00172 *> \endverbatim
00173 *>
00174 *> \param[out] WORK
00175 *> \verbatim
00176 *>          WORK is REAL array, dimension LWORK
00177 *>           On exit, if LWORK = -1, WORK(1) returns an estimate of
00178 *>           the optimal value for LWORK.
00179 *> \endverbatim
00180 *>
00181 *> \param[in] LWORK
00182 *> \verbatim
00183 *>          LWORK is INTEGER
00184 *>           The dimension of the array WORK.  LWORK .GE. max(1,N)
00185 *>           is sufficient, but LWORK typically as large as 6*N may
00186 *>           be required for optimal performance.  A workspace query
00187 *>           to determine the optimal workspace size is recommended.
00188 *>
00189 *>           If LWORK = -1, then SLAQR4 does a workspace query.
00190 *>           In this case, SLAQR4 checks the input parameters and
00191 *>           estimates the optimal workspace size for the given
00192 *>           values of N, ILO and IHI.  The estimate is returned
00193 *>           in WORK(1).  No error message related to LWORK is
00194 *>           issued by XERBLA.  Neither H nor Z are accessed.
00195 *> \endverbatim
00196 *>
00197 *> \param[out] INFO
00198 *> \verbatim
00199 *>          INFO is INTEGER
00200 *> \verbatim
00201 *>          INFO is INTEGER
00202 *>             =  0:  successful exit
00203 *>           .GT. 0:  if INFO = i, SLAQR4 failed to compute all of
00204 *>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00205 *>                and WI contain those eigenvalues which have been
00206 *>                successfully computed.  (Failures are rare.)
00207 *>
00208 *>                If INFO .GT. 0 and WANT is .FALSE., then on exit,
00209 *>                the remaining unconverged eigenvalues are the eigen-
00210 *>                values of the upper Hessenberg matrix rows and
00211 *>                columns ILO through INFO of the final, output
00212 *>                value of H.
00213 *>
00214 *>                If INFO .GT. 0 and WANTT is .TRUE., then on exit
00215 *>
00216 *>           (*)  (initial value of H)*U  = U*(final value of H)
00217 *>
00218 *>                where U is a orthogonal matrix.  The final
00219 *>                value of  H is upper Hessenberg and triangular in
00220 *>                rows and columns INFO+1 through IHI.
00221 *>
00222 *>                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
00223 *>
00224 *>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
00225 *>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
00226 *>
00227 *>                where U is the orthogonal matrix in (*) (regard-
00228 *>                less of the value of WANTT.)
00229 *>
00230 *>                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
00231 *>                accessed.
00232 *> \endverbatim
00233 *
00234 *  Authors:
00235 *  ========
00236 *
00237 *> \author Univ. of Tennessee 
00238 *> \author Univ. of California Berkeley 
00239 *> \author Univ. of Colorado Denver 
00240 *> \author NAG Ltd. 
00241 *
00242 *> \date November 2011
00243 *
00244 *> \ingroup realOTHERauxiliary
00245 *
00246 *> \par Contributors:
00247 *  ==================
00248 *>
00249 *>       Karen Braman and Ralph Byers, Department of Mathematics,
00250 *>       University of Kansas, USA
00251 *
00252 *> \par References:
00253 *  ================
00254 *>
00255 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00256 *>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00257 *>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00258 *>       929--947, 2002.
00259 *> \n
00260 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00261 *>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00262 *>       of Matrix Analysis, volume 23, pages 948--973, 2002.
00263 *>
00264 *  =====================================================================
00265       SUBROUTINE SLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00266      $                   ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
00267 *
00268 *  -- LAPACK auxiliary routine (version 3.4.0) --
00269 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00270 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00271 *     November 2011
00272 *
00273 *     .. Scalar Arguments ..
00274       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
00275       LOGICAL            WANTT, WANTZ
00276 *     ..
00277 *     .. Array Arguments ..
00278       REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
00279      $                   Z( LDZ, * )
00280 *     ..
00281 *
00282 *  ================================================================
00283 *
00284 *     .. Parameters ..
00285 *
00286 *     ==== Matrices of order NTINY or smaller must be processed by
00287 *     .    SLAHQR because of insufficient subdiagonal scratch space.
00288 *     .    (This is a hard limit.) ====
00289       INTEGER            NTINY
00290       PARAMETER          ( NTINY = 11 )
00291 *
00292 *     ==== Exceptional deflation windows:  try to cure rare
00293 *     .    slow convergence by varying the size of the
00294 *     .    deflation window after KEXNW iterations. ====
00295       INTEGER            KEXNW
00296       PARAMETER          ( KEXNW = 5 )
00297 *
00298 *     ==== Exceptional shifts: try to cure rare slow convergence
00299 *     .    with ad-hoc exceptional shifts every KEXSH iterations.
00300 *     .    ====
00301       INTEGER            KEXSH
00302       PARAMETER          ( KEXSH = 6 )
00303 *
00304 *     ==== The constants WILK1 and WILK2 are used to form the
00305 *     .    exceptional shifts. ====
00306       REAL               WILK1, WILK2
00307       PARAMETER          ( WILK1 = 0.75e0, WILK2 = -0.4375e0 )
00308       REAL               ZERO, ONE
00309       PARAMETER          ( ZERO = 0.0e0, ONE = 1.0e0 )
00310 *     ..
00311 *     .. Local Scalars ..
00312       REAL               AA, BB, CC, CS, DD, SN, SS, SWAP
00313       INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
00314      $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
00315      $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
00316      $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
00317       LOGICAL            SORTED
00318       CHARACTER          JBCMPZ*2
00319 *     ..
00320 *     .. External Functions ..
00321       INTEGER            ILAENV
00322       EXTERNAL           ILAENV
00323 *     ..
00324 *     .. Local Arrays ..
00325       REAL               ZDUM( 1, 1 )
00326 *     ..
00327 *     .. External Subroutines ..
00328       EXTERNAL           SLACPY, SLAHQR, SLANV2, SLAQR2, SLAQR5
00329 *     ..
00330 *     .. Intrinsic Functions ..
00331       INTRINSIC          ABS, INT, MAX, MIN, MOD, REAL
00332 *     ..
00333 *     .. Executable Statements ..
00334       INFO = 0
00335 *
00336 *     ==== Quick return for N = 0: nothing to do. ====
00337 *
00338       IF( N.EQ.0 ) THEN
00339          WORK( 1 ) = ONE
00340          RETURN
00341       END IF
00342 *
00343       IF( N.LE.NTINY ) THEN
00344 *
00345 *        ==== Tiny matrices must use SLAHQR. ====
00346 *
00347          LWKOPT = 1
00348          IF( LWORK.NE.-1 )
00349      $      CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00350      $                   ILOZ, IHIZ, Z, LDZ, INFO )
00351       ELSE
00352 *
00353 *        ==== Use small bulge multi-shift QR with aggressive early
00354 *        .    deflation on larger-than-tiny matrices. ====
00355 *
00356 *        ==== Hope for the best. ====
00357 *
00358          INFO = 0
00359 *
00360 *        ==== Set up job flags for ILAENV. ====
00361 *
00362          IF( WANTT ) THEN
00363             JBCMPZ( 1: 1 ) = 'S'
00364          ELSE
00365             JBCMPZ( 1: 1 ) = 'E'
00366          END IF
00367          IF( WANTZ ) THEN
00368             JBCMPZ( 2: 2 ) = 'V'
00369          ELSE
00370             JBCMPZ( 2: 2 ) = 'N'
00371          END IF
00372 *
00373 *        ==== NWR = recommended deflation window size.  At this
00374 *        .    point,  N .GT. NTINY = 11, so there is enough
00375 *        .    subdiagonal workspace for NWR.GE.2 as required.
00376 *        .    (In fact, there is enough subdiagonal space for
00377 *        .    NWR.GE.3.) ====
00378 *
00379          NWR = ILAENV( 13, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00380          NWR = MAX( 2, NWR )
00381          NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
00382 *
00383 *        ==== NSR = recommended number of simultaneous shifts.
00384 *        .    At this point N .GT. NTINY = 11, so there is at
00385 *        .    enough subdiagonal workspace for NSR to be even
00386 *        .    and greater than or equal to two as required. ====
00387 *
00388          NSR = ILAENV( 15, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00389          NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
00390          NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
00391 *
00392 *        ==== Estimate optimal workspace ====
00393 *
00394 *        ==== Workspace query call to SLAQR2 ====
00395 *
00396          CALL SLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
00397      $                IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
00398      $                N, H, LDH, WORK, -1 )
00399 *
00400 *        ==== Optimal workspace = MAX(SLAQR5, SLAQR2) ====
00401 *
00402          LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
00403 *
00404 *        ==== Quick return in case of workspace query. ====
00405 *
00406          IF( LWORK.EQ.-1 ) THEN
00407             WORK( 1 ) = REAL( LWKOPT )
00408             RETURN
00409          END IF
00410 *
00411 *        ==== SLAHQR/SLAQR0 crossover point ====
00412 *
00413          NMIN = ILAENV( 12, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00414          NMIN = MAX( NTINY, NMIN )
00415 *
00416 *        ==== Nibble crossover point ====
00417 *
00418          NIBBLE = ILAENV( 14, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00419          NIBBLE = MAX( 0, NIBBLE )
00420 *
00421 *        ==== Accumulate reflections during ttswp?  Use block
00422 *        .    2-by-2 structure during matrix-matrix multiply? ====
00423 *
00424          KACC22 = ILAENV( 16, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00425          KACC22 = MAX( 0, KACC22 )
00426          KACC22 = MIN( 2, KACC22 )
00427 *
00428 *        ==== NWMAX = the largest possible deflation window for
00429 *        .    which there is sufficient workspace. ====
00430 *
00431          NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
00432          NW = NWMAX
00433 *
00434 *        ==== NSMAX = the Largest number of simultaneous shifts
00435 *        .    for which there is sufficient workspace. ====
00436 *
00437          NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
00438          NSMAX = NSMAX - MOD( NSMAX, 2 )
00439 *
00440 *        ==== NDFL: an iteration count restarted at deflation. ====
00441 *
00442          NDFL = 1
00443 *
00444 *        ==== ITMAX = iteration limit ====
00445 *
00446          ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
00447 *
00448 *        ==== Last row and column in the active block ====
00449 *
00450          KBOT = IHI
00451 *
00452 *        ==== Main Loop ====
00453 *
00454          DO 80 IT = 1, ITMAX
00455 *
00456 *           ==== Done when KBOT falls below ILO ====
00457 *
00458             IF( KBOT.LT.ILO )
00459      $         GO TO 90
00460 *
00461 *           ==== Locate active block ====
00462 *
00463             DO 10 K = KBOT, ILO + 1, -1
00464                IF( H( K, K-1 ).EQ.ZERO )
00465      $            GO TO 20
00466    10       CONTINUE
00467             K = ILO
00468    20       CONTINUE
00469             KTOP = K
00470 *
00471 *           ==== Select deflation window size:
00472 *           .    Typical Case:
00473 *           .      If possible and advisable, nibble the entire
00474 *           .      active block.  If not, use size MIN(NWR,NWMAX)
00475 *           .      or MIN(NWR+1,NWMAX) depending upon which has
00476 *           .      the smaller corresponding subdiagonal entry
00477 *           .      (a heuristic).
00478 *           .
00479 *           .    Exceptional Case:
00480 *           .      If there have been no deflations in KEXNW or
00481 *           .      more iterations, then vary the deflation window
00482 *           .      size.   At first, because, larger windows are,
00483 *           .      in general, more powerful than smaller ones,
00484 *           .      rapidly increase the window to the maximum possible.
00485 *           .      Then, gradually reduce the window size. ====
00486 *
00487             NH = KBOT - KTOP + 1
00488             NWUPBD = MIN( NH, NWMAX )
00489             IF( NDFL.LT.KEXNW ) THEN
00490                NW = MIN( NWUPBD, NWR )
00491             ELSE
00492                NW = MIN( NWUPBD, 2*NW )
00493             END IF
00494             IF( NW.LT.NWMAX ) THEN
00495                IF( NW.GE.NH-1 ) THEN
00496                   NW = NH
00497                ELSE
00498                   KWTOP = KBOT - NW + 1
00499                   IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
00500      $                ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
00501                END IF
00502             END IF
00503             IF( NDFL.LT.KEXNW ) THEN
00504                NDEC = -1
00505             ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
00506                NDEC = NDEC + 1
00507                IF( NW-NDEC.LT.2 )
00508      $            NDEC = 0
00509                NW = NW - NDEC
00510             END IF
00511 *
00512 *           ==== Aggressive early deflation:
00513 *           .    split workspace under the subdiagonal into
00514 *           .      - an nw-by-nw work array V in the lower
00515 *           .        left-hand-corner,
00516 *           .      - an NW-by-at-least-NW-but-more-is-better
00517 *           .        (NW-by-NHO) horizontal work array along
00518 *           .        the bottom edge,
00519 *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
00520 *           .        vertical work array along the left-hand-edge.
00521 *           .        ====
00522 *
00523             KV = N - NW + 1
00524             KT = NW + 1
00525             NHO = ( N-NW-1 ) - KT + 1
00526             KWV = NW + 2
00527             NVE = ( N-NW ) - KWV + 1
00528 *
00529 *           ==== Aggressive early deflation ====
00530 *
00531             CALL SLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
00532      $                   IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
00533      $                   NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
00534      $                   WORK, LWORK )
00535 *
00536 *           ==== Adjust KBOT accounting for new deflations. ====
00537 *
00538             KBOT = KBOT - LD
00539 *
00540 *           ==== KS points to the shifts. ====
00541 *
00542             KS = KBOT - LS + 1
00543 *
00544 *           ==== Skip an expensive QR sweep if there is a (partly
00545 *           .    heuristic) reason to expect that many eigenvalues
00546 *           .    will deflate without it.  Here, the QR sweep is
00547 *           .    skipped if many eigenvalues have just been deflated
00548 *           .    or if the remaining active block is small.
00549 *
00550             IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
00551      $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
00552 *
00553 *              ==== NS = nominal number of simultaneous shifts.
00554 *              .    This may be lowered (slightly) if SLAQR2
00555 *              .    did not provide that many shifts. ====
00556 *
00557                NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
00558                NS = NS - MOD( NS, 2 )
00559 *
00560 *              ==== If there have been no deflations
00561 *              .    in a multiple of KEXSH iterations,
00562 *              .    then try exceptional shifts.
00563 *              .    Otherwise use shifts provided by
00564 *              .    SLAQR2 above or from the eigenvalues
00565 *              .    of a trailing principal submatrix. ====
00566 *
00567                IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
00568                   KS = KBOT - NS + 1
00569                   DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
00570                      SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
00571                      AA = WILK1*SS + H( I, I )
00572                      BB = SS
00573                      CC = WILK2*SS
00574                      DD = AA
00575                      CALL SLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
00576      $                            WR( I ), WI( I ), CS, SN )
00577    30             CONTINUE
00578                   IF( KS.EQ.KTOP ) THEN
00579                      WR( KS+1 ) = H( KS+1, KS+1 )
00580                      WI( KS+1 ) = ZERO
00581                      WR( KS ) = WR( KS+1 )
00582                      WI( KS ) = WI( KS+1 )
00583                   END IF
00584                ELSE
00585 *
00586 *                 ==== Got NS/2 or fewer shifts? Use SLAHQR
00587 *                 .    on a trailing principal submatrix to
00588 *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
00589 *                 .    there is enough space below the subdiagonal
00590 *                 .    to fit an NS-by-NS scratch array.) ====
00591 *
00592                   IF( KBOT-KS+1.LE.NS / 2 ) THEN
00593                      KS = KBOT - NS + 1
00594                      KT = N - NS + 1
00595                      CALL SLACPY( 'A', NS, NS, H( KS, KS ), LDH,
00596      $                            H( KT, 1 ), LDH )
00597                      CALL SLAHQR( .false., .false., NS, 1, NS,
00598      $                            H( KT, 1 ), LDH, WR( KS ), WI( KS ),
00599      $                            1, 1, ZDUM, 1, INF )
00600                      KS = KS + INF
00601 *
00602 *                    ==== In case of a rare QR failure use
00603 *                    .    eigenvalues of the trailing 2-by-2
00604 *                    .    principal submatrix.  ====
00605 *
00606                      IF( KS.GE.KBOT ) THEN
00607                         AA = H( KBOT-1, KBOT-1 )
00608                         CC = H( KBOT, KBOT-1 )
00609                         BB = H( KBOT-1, KBOT )
00610                         DD = H( KBOT, KBOT )
00611                         CALL SLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
00612      $                               WI( KBOT-1 ), WR( KBOT ),
00613      $                               WI( KBOT ), CS, SN )
00614                         KS = KBOT - 1
00615                      END IF
00616                   END IF
00617 *
00618                   IF( KBOT-KS+1.GT.NS ) THEN
00619 *
00620 *                    ==== Sort the shifts (Helps a little)
00621 *                    .    Bubble sort keeps complex conjugate
00622 *                    .    pairs together. ====
00623 *
00624                      SORTED = .false.
00625                      DO 50 K = KBOT, KS + 1, -1
00626                         IF( SORTED )
00627      $                     GO TO 60
00628                         SORTED = .true.
00629                         DO 40 I = KS, K - 1
00630                            IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
00631      $                         ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
00632                               SORTED = .false.
00633 *
00634                               SWAP = WR( I )
00635                               WR( I ) = WR( I+1 )
00636                               WR( I+1 ) = SWAP
00637 *
00638                               SWAP = WI( I )
00639                               WI( I ) = WI( I+1 )
00640                               WI( I+1 ) = SWAP
00641                            END IF
00642    40                   CONTINUE
00643    50                CONTINUE
00644    60                CONTINUE
00645                   END IF
00646 *
00647 *                 ==== Shuffle shifts into pairs of real shifts
00648 *                 .    and pairs of complex conjugate shifts
00649 *                 .    assuming complex conjugate shifts are
00650 *                 .    already adjacent to one another. (Yes,
00651 *                 .    they are.)  ====
00652 *
00653                   DO 70 I = KBOT, KS + 2, -2
00654                      IF( WI( I ).NE.-WI( I-1 ) ) THEN
00655 *
00656                         SWAP = WR( I )
00657                         WR( I ) = WR( I-1 )
00658                         WR( I-1 ) = WR( I-2 )
00659                         WR( I-2 ) = SWAP
00660 *
00661                         SWAP = WI( I )
00662                         WI( I ) = WI( I-1 )
00663                         WI( I-1 ) = WI( I-2 )
00664                         WI( I-2 ) = SWAP
00665                      END IF
00666    70             CONTINUE
00667                END IF
00668 *
00669 *              ==== If there are only two shifts and both are
00670 *              .    real, then use only one.  ====
00671 *
00672                IF( KBOT-KS+1.EQ.2 ) THEN
00673                   IF( WI( KBOT ).EQ.ZERO ) THEN
00674                      IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
00675      $                   ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
00676                         WR( KBOT-1 ) = WR( KBOT )
00677                      ELSE
00678                         WR( KBOT ) = WR( KBOT-1 )
00679                      END IF
00680                   END IF
00681                END IF
00682 *
00683 *              ==== Use up to NS of the the smallest magnatiude
00684 *              .    shifts.  If there aren't NS shifts available,
00685 *              .    then use them all, possibly dropping one to
00686 *              .    make the number of shifts even. ====
00687 *
00688                NS = MIN( NS, KBOT-KS+1 )
00689                NS = NS - MOD( NS, 2 )
00690                KS = KBOT - NS + 1
00691 *
00692 *              ==== Small-bulge multi-shift QR sweep:
00693 *              .    split workspace under the subdiagonal into
00694 *              .    - a KDU-by-KDU work array U in the lower
00695 *              .      left-hand-corner,
00696 *              .    - a KDU-by-at-least-KDU-but-more-is-better
00697 *              .      (KDU-by-NHo) horizontal work array WH along
00698 *              .      the bottom edge,
00699 *              .    - and an at-least-KDU-but-more-is-better-by-KDU
00700 *              .      (NVE-by-KDU) vertical work WV arrow along
00701 *              .      the left-hand-edge. ====
00702 *
00703                KDU = 3*NS - 3
00704                KU = N - KDU + 1
00705                KWH = KDU + 1
00706                NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
00707                KWV = KDU + 4
00708                NVE = N - KDU - KWV + 1
00709 *
00710 *              ==== Small-bulge multi-shift QR sweep ====
00711 *
00712                CALL SLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
00713      $                      WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
00714      $                      LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
00715      $                      H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
00716             END IF
00717 *
00718 *           ==== Note progress (or the lack of it). ====
00719 *
00720             IF( LD.GT.0 ) THEN
00721                NDFL = 1
00722             ELSE
00723                NDFL = NDFL + 1
00724             END IF
00725 *
00726 *           ==== End of main loop ====
00727    80    CONTINUE
00728 *
00729 *        ==== Iteration limit exceeded.  Set INFO to show where
00730 *        .    the problem occurred and exit. ====
00731 *
00732          INFO = KBOT
00733    90    CONTINUE
00734       END IF
00735 *
00736 *     ==== Return the optimal value of LWORK. ====
00737 *
00738       WORK( 1 ) = REAL( LWKOPT )
00739 *
00740 *     ==== End of SLAQR4 ====
00741 *
00742       END
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