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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZLAQR0 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZLAQR0 + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr0.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr0.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr0.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZLAQR0( 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 *> ZLAQR0 computes the eigenvalues of a Hessenberg matrix H 00039 *> and, optionally, the matrices T and Z from the Schur decomposition 00040 *> H = Z T Z**H, where T is an upper triangular matrix (the 00041 *> Schur form), and Z is the unitary matrix of Schur vectors. 00042 *> 00043 *> Optionally Z may be postmultiplied into an input unitary 00044 *> matrix Q so that this routine can give the Schur factorization 00045 *> of a matrix A which has been reduced to the Hessenberg form H 00046 *> by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. 00047 *> \endverbatim 00048 * 00049 * Arguments: 00050 * ========== 00051 * 00052 *> \param[in] WANTT 00053 *> \verbatim 00054 *> WANTT is LOGICAL 00055 *> = .TRUE. : the full Schur form T is required; 00056 *> = .FALSE.: only eigenvalues are required. 00057 *> \endverbatim 00058 *> 00059 *> \param[in] WANTZ 00060 *> \verbatim 00061 *> WANTZ is LOGICAL 00062 *> = .TRUE. : the matrix of Schur vectors Z is required; 00063 *> = .FALSE.: Schur vectors are not required. 00064 *> \endverbatim 00065 *> 00066 *> \param[in] N 00067 *> \verbatim 00068 *> N is INTEGER 00069 *> The order of the matrix H. N .GE. 0. 00070 *> \endverbatim 00071 *> 00072 *> \param[in] ILO 00073 *> \verbatim 00074 *> ILO is INTEGER 00075 *> \endverbatim 00076 *> 00077 *> \param[in] IHI 00078 *> \verbatim 00079 *> IHI is INTEGER 00080 *> 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 ZGEBAL, and then passed to ZGEHRD when the 00085 *> matrix output by ZGEBAL 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 COMPLEX*16 array, dimension (LDH,N) 00094 *> On entry, the upper Hessenberg matrix H. 00095 *> On exit, if INFO = 0 and WANTT is .TRUE., then H 00096 *> contains the upper triangular matrix T from the Schur 00097 *> decomposition (the Schur form). If INFO = 0 and WANT is 00098 *> .FALSE., then the contents of H are unspecified on exit. 00099 *> (The output value of H when INFO.GT.0 is given under the 00100 *> description of INFO below.) 00101 *> 00102 *> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and 00103 *> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDH 00107 *> \verbatim 00108 *> LDH is INTEGER 00109 *> The leading dimension of the array H. LDH .GE. max(1,N). 00110 *> \endverbatim 00111 *> 00112 *> \param[out] W 00113 *> \verbatim 00114 *> W is COMPLEX*16 array, dimension (N) 00115 *> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored 00116 *> in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are 00117 *> stored in the same order as on the diagonal of the Schur 00118 *> form returned in H, with W(i) = H(i,i). 00119 *> \endverbatim 00120 *> 00121 *> \param[in] ILOZ 00122 *> \verbatim 00123 *> ILOZ is INTEGER 00124 *> \endverbatim 00125 *> 00126 *> \param[in] IHIZ 00127 *> \verbatim 00128 *> IHIZ is INTEGER 00129 *> Specify the rows of Z to which transformations must be 00130 *> applied if WANTZ is .TRUE.. 00131 *> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. 00132 *> \endverbatim 00133 *> 00134 *> \param[in,out] Z 00135 *> \verbatim 00136 *> Z is COMPLEX*16 array, dimension (LDZ,IHI) 00137 *> If WANTZ is .FALSE., then Z is not referenced. 00138 *> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is 00139 *> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the 00140 *> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). 00141 *> (The output value of Z when INFO.GT.0 is given under 00142 *> the description of INFO below.) 00143 *> \endverbatim 00144 *> 00145 *> \param[in] LDZ 00146 *> \verbatim 00147 *> LDZ is INTEGER 00148 *> The leading dimension of the array Z. if WANTZ is .TRUE. 00149 *> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. 00150 *> \endverbatim 00151 *> 00152 *> \param[out] WORK 00153 *> \verbatim 00154 *> WORK is COMPLEX*16 array, dimension LWORK 00155 *> On exit, if LWORK = -1, WORK(1) returns an estimate of 00156 *> the optimal value for LWORK. 00157 *> \endverbatim 00158 *> 00159 *> \param[in] LWORK 00160 *> \verbatim 00161 *> LWORK is INTEGER 00162 *> The dimension of the array WORK. LWORK .GE. max(1,N) 00163 *> is sufficient, but LWORK typically as large as 6*N may 00164 *> be required for optimal performance. A workspace query 00165 *> to determine the optimal workspace size is recommended. 00166 *> 00167 *> If LWORK = -1, then ZLAQR0 does a workspace query. 00168 *> In this case, ZLAQR0 checks the input parameters and 00169 *> estimates the optimal workspace size for the given 00170 *> values of N, ILO and IHI. The estimate is returned 00171 *> in WORK(1). No error message related to LWORK is 00172 *> issued by XERBLA. Neither H nor Z are accessed. 00173 *> \endverbatim 00174 *> 00175 *> \param[out] INFO 00176 *> \verbatim 00177 *> INFO is INTEGER 00178 *> = 0: successful exit 00179 *> .GT. 0: if INFO = i, ZLAQR0 failed to compute all of 00180 *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR 00181 *> and WI contain those eigenvalues which have been 00182 *> successfully computed. (Failures are rare.) 00183 *> 00184 *> If INFO .GT. 0 and WANT is .FALSE., then on exit, 00185 *> the remaining unconverged eigenvalues are the eigen- 00186 *> values of the upper Hessenberg matrix rows and 00187 *> columns ILO through INFO of the final, output 00188 *> value of H. 00189 *> 00190 *> If INFO .GT. 0 and WANTT is .TRUE., then on exit 00191 *> 00192 *> (*) (initial value of H)*U = U*(final value of H) 00193 *> 00194 *> where U is a unitary matrix. The final 00195 *> value of H is upper Hessenberg and triangular in 00196 *> rows and columns INFO+1 through IHI. 00197 *> 00198 *> If INFO .GT. 0 and WANTZ is .TRUE., then on exit 00199 *> 00200 *> (final value of Z(ILO:IHI,ILOZ:IHIZ) 00201 *> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U 00202 *> 00203 *> where U is the unitary matrix in (*) (regard- 00204 *> less of the value of WANTT.) 00205 *> 00206 *> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not 00207 *> accessed. 00208 *> \endverbatim 00209 * 00210 * Authors: 00211 * ======== 00212 * 00213 *> \author Univ. of Tennessee 00214 *> \author Univ. of California Berkeley 00215 *> \author Univ. of Colorado Denver 00216 *> \author NAG Ltd. 00217 * 00218 *> \date November 2011 00219 * 00220 *> \ingroup complex16OTHERauxiliary 00221 * 00222 *> \par Contributors: 00223 * ================== 00224 *> 00225 *> Karen Braman and Ralph Byers, Department of Mathematics, 00226 *> University of Kansas, USA 00227 * 00228 *> \par References: 00229 * ================ 00230 *> 00231 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 00232 *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 00233 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages 00234 *> 929--947, 2002. 00235 *> \n 00236 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 00237 *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal 00238 *> of Matrix Analysis, volume 23, pages 948--973, 2002. 00239 *> 00240 * ===================================================================== 00241 SUBROUTINE ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 00242 $ IHIZ, Z, LDZ, WORK, LWORK, INFO ) 00243 * 00244 * -- LAPACK auxiliary routine (version 3.4.0) -- 00245 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00246 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00247 * November 2011 00248 * 00249 * .. Scalar Arguments .. 00250 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N 00251 LOGICAL WANTT, WANTZ 00252 * .. 00253 * .. Array Arguments .. 00254 COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) 00255 * .. 00256 * 00257 * ================================================================ 00258 * 00259 * .. Parameters .. 00260 * 00261 * ==== Matrices of order NTINY or smaller must be processed by 00262 * . ZLAHQR because of insufficient subdiagonal scratch space. 00263 * . (This is a hard limit.) ==== 00264 INTEGER NTINY 00265 PARAMETER ( NTINY = 11 ) 00266 * 00267 * ==== Exceptional deflation windows: try to cure rare 00268 * . slow convergence by varying the size of the 00269 * . deflation window after KEXNW iterations. ==== 00270 INTEGER KEXNW 00271 PARAMETER ( KEXNW = 5 ) 00272 * 00273 * ==== Exceptional shifts: try to cure rare slow convergence 00274 * . with ad-hoc exceptional shifts every KEXSH iterations. 00275 * . ==== 00276 INTEGER KEXSH 00277 PARAMETER ( KEXSH = 6 ) 00278 * 00279 * ==== The constant WILK1 is used to form the exceptional 00280 * . shifts. ==== 00281 DOUBLE PRECISION WILK1 00282 PARAMETER ( WILK1 = 0.75d0 ) 00283 COMPLEX*16 ZERO, ONE 00284 PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), 00285 $ ONE = ( 1.0d0, 0.0d0 ) ) 00286 DOUBLE PRECISION TWO 00287 PARAMETER ( TWO = 2.0d0 ) 00288 * .. 00289 * .. Local Scalars .. 00290 COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2 00291 DOUBLE PRECISION S 00292 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, 00293 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, 00294 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS, 00295 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD 00296 LOGICAL SORTED 00297 CHARACTER JBCMPZ*2 00298 * .. 00299 * .. External Functions .. 00300 INTEGER ILAENV 00301 EXTERNAL ILAENV 00302 * .. 00303 * .. Local Arrays .. 00304 COMPLEX*16 ZDUM( 1, 1 ) 00305 * .. 00306 * .. External Subroutines .. 00307 EXTERNAL ZLACPY, ZLAHQR, ZLAQR3, ZLAQR4, ZLAQR5 00308 * .. 00309 * .. Intrinsic Functions .. 00310 INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD, 00311 $ SQRT 00312 * .. 00313 * .. Statement Functions .. 00314 DOUBLE PRECISION CABS1 00315 * .. 00316 * .. Statement Function definitions .. 00317 CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) ) 00318 * .. 00319 * .. Executable Statements .. 00320 INFO = 0 00321 * 00322 * ==== Quick return for N = 0: nothing to do. ==== 00323 * 00324 IF( N.EQ.0 ) THEN 00325 WORK( 1 ) = ONE 00326 RETURN 00327 END IF 00328 * 00329 IF( N.LE.NTINY ) THEN 00330 * 00331 * ==== Tiny matrices must use ZLAHQR. ==== 00332 * 00333 LWKOPT = 1 00334 IF( LWORK.NE.-1 ) 00335 $ CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 00336 $ IHIZ, Z, LDZ, INFO ) 00337 ELSE 00338 * 00339 * ==== Use small bulge multi-shift QR with aggressive early 00340 * . deflation on larger-than-tiny matrices. ==== 00341 * 00342 * ==== Hope for the best. ==== 00343 * 00344 INFO = 0 00345 * 00346 * ==== Set up job flags for ILAENV. ==== 00347 * 00348 IF( WANTT ) THEN 00349 JBCMPZ( 1: 1 ) = 'S' 00350 ELSE 00351 JBCMPZ( 1: 1 ) = 'E' 00352 END IF 00353 IF( WANTZ ) THEN 00354 JBCMPZ( 2: 2 ) = 'V' 00355 ELSE 00356 JBCMPZ( 2: 2 ) = 'N' 00357 END IF 00358 * 00359 * ==== NWR = recommended deflation window size. At this 00360 * . point, N .GT. NTINY = 11, so there is enough 00361 * . subdiagonal workspace for NWR.GE.2 as required. 00362 * . (In fact, there is enough subdiagonal space for 00363 * . NWR.GE.3.) ==== 00364 * 00365 NWR = ILAENV( 13, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00366 NWR = MAX( 2, NWR ) 00367 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) 00368 * 00369 * ==== NSR = recommended number of simultaneous shifts. 00370 * . At this point N .GT. NTINY = 11, so there is at 00371 * . enough subdiagonal workspace for NSR to be even 00372 * . and greater than or equal to two as required. ==== 00373 * 00374 NSR = ILAENV( 15, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00375 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO ) 00376 NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) 00377 * 00378 * ==== Estimate optimal workspace ==== 00379 * 00380 * ==== Workspace query call to ZLAQR3 ==== 00381 * 00382 CALL ZLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, 00383 $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H, 00384 $ LDH, WORK, -1 ) 00385 * 00386 * ==== Optimal workspace = MAX(ZLAQR5, ZLAQR3) ==== 00387 * 00388 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) 00389 * 00390 * ==== Quick return in case of workspace query. ==== 00391 * 00392 IF( LWORK.EQ.-1 ) THEN 00393 WORK( 1 ) = DCMPLX( LWKOPT, 0 ) 00394 RETURN 00395 END IF 00396 * 00397 * ==== ZLAHQR/ZLAQR0 crossover point ==== 00398 * 00399 NMIN = ILAENV( 12, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00400 NMIN = MAX( NTINY, NMIN ) 00401 * 00402 * ==== Nibble crossover point ==== 00403 * 00404 NIBBLE = ILAENV( 14, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00405 NIBBLE = MAX( 0, NIBBLE ) 00406 * 00407 * ==== Accumulate reflections during ttswp? Use block 00408 * . 2-by-2 structure during matrix-matrix multiply? ==== 00409 * 00410 KACC22 = ILAENV( 16, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00411 KACC22 = MAX( 0, KACC22 ) 00412 KACC22 = MIN( 2, KACC22 ) 00413 * 00414 * ==== NWMAX = the largest possible deflation window for 00415 * . which there is sufficient workspace. ==== 00416 * 00417 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) 00418 NW = NWMAX 00419 * 00420 * ==== NSMAX = the Largest number of simultaneous shifts 00421 * . for which there is sufficient workspace. ==== 00422 * 00423 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 ) 00424 NSMAX = NSMAX - MOD( NSMAX, 2 ) 00425 * 00426 * ==== NDFL: an iteration count restarted at deflation. ==== 00427 * 00428 NDFL = 1 00429 * 00430 * ==== ITMAX = iteration limit ==== 00431 * 00432 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) 00433 * 00434 * ==== Last row and column in the active block ==== 00435 * 00436 KBOT = IHI 00437 * 00438 * ==== Main Loop ==== 00439 * 00440 DO 70 IT = 1, ITMAX 00441 * 00442 * ==== Done when KBOT falls below ILO ==== 00443 * 00444 IF( KBOT.LT.ILO ) 00445 $ GO TO 80 00446 * 00447 * ==== Locate active block ==== 00448 * 00449 DO 10 K = KBOT, ILO + 1, -1 00450 IF( H( K, K-1 ).EQ.ZERO ) 00451 $ GO TO 20 00452 10 CONTINUE 00453 K = ILO 00454 20 CONTINUE 00455 KTOP = K 00456 * 00457 * ==== Select deflation window size: 00458 * . Typical Case: 00459 * . If possible and advisable, nibble the entire 00460 * . active block. If not, use size MIN(NWR,NWMAX) 00461 * . or MIN(NWR+1,NWMAX) depending upon which has 00462 * . the smaller corresponding subdiagonal entry 00463 * . (a heuristic). 00464 * . 00465 * . Exceptional Case: 00466 * . If there have been no deflations in KEXNW or 00467 * . more iterations, then vary the deflation window 00468 * . size. At first, because, larger windows are, 00469 * . in general, more powerful than smaller ones, 00470 * . rapidly increase the window to the maximum possible. 00471 * . Then, gradually reduce the window size. ==== 00472 * 00473 NH = KBOT - KTOP + 1 00474 NWUPBD = MIN( NH, NWMAX ) 00475 IF( NDFL.LT.KEXNW ) THEN 00476 NW = MIN( NWUPBD, NWR ) 00477 ELSE 00478 NW = MIN( NWUPBD, 2*NW ) 00479 END IF 00480 IF( NW.LT.NWMAX ) THEN 00481 IF( NW.GE.NH-1 ) THEN 00482 NW = NH 00483 ELSE 00484 KWTOP = KBOT - NW + 1 00485 IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT. 00486 $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 00487 END IF 00488 END IF 00489 IF( NDFL.LT.KEXNW ) THEN 00490 NDEC = -1 00491 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN 00492 NDEC = NDEC + 1 00493 IF( NW-NDEC.LT.2 ) 00494 $ NDEC = 0 00495 NW = NW - NDEC 00496 END IF 00497 * 00498 * ==== Aggressive early deflation: 00499 * . split workspace under the subdiagonal into 00500 * . - an nw-by-nw work array V in the lower 00501 * . left-hand-corner, 00502 * . - an NW-by-at-least-NW-but-more-is-better 00503 * . (NW-by-NHO) horizontal work array along 00504 * . the bottom edge, 00505 * . - an at-least-NW-but-more-is-better (NHV-by-NW) 00506 * . vertical work array along the left-hand-edge. 00507 * . ==== 00508 * 00509 KV = N - NW + 1 00510 KT = NW + 1 00511 NHO = ( N-NW-1 ) - KT + 1 00512 KWV = NW + 2 00513 NVE = ( N-NW ) - KWV + 1 00514 * 00515 * ==== Aggressive early deflation ==== 00516 * 00517 CALL ZLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, 00518 $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO, 00519 $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK, 00520 $ LWORK ) 00521 * 00522 * ==== Adjust KBOT accounting for new deflations. ==== 00523 * 00524 KBOT = KBOT - LD 00525 * 00526 * ==== KS points to the shifts. ==== 00527 * 00528 KS = KBOT - LS + 1 00529 * 00530 * ==== Skip an expensive QR sweep if there is a (partly 00531 * . heuristic) reason to expect that many eigenvalues 00532 * . will deflate without it. Here, the QR sweep is 00533 * . skipped if many eigenvalues have just been deflated 00534 * . or if the remaining active block is small. 00535 * 00536 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- 00537 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN 00538 * 00539 * ==== NS = nominal number of simultaneous shifts. 00540 * . This may be lowered (slightly) if ZLAQR3 00541 * . did not provide that many shifts. ==== 00542 * 00543 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) 00544 NS = NS - MOD( NS, 2 ) 00545 * 00546 * ==== If there have been no deflations 00547 * . in a multiple of KEXSH iterations, 00548 * . then try exceptional shifts. 00549 * . Otherwise use shifts provided by 00550 * . ZLAQR3 above or from the eigenvalues 00551 * . of a trailing principal submatrix. ==== 00552 * 00553 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN 00554 KS = KBOT - NS + 1 00555 DO 30 I = KBOT, KS + 1, -2 00556 W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) ) 00557 W( I-1 ) = W( I ) 00558 30 CONTINUE 00559 ELSE 00560 * 00561 * ==== Got NS/2 or fewer shifts? Use ZLAQR4 or 00562 * . ZLAHQR on a trailing principal submatrix to 00563 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, 00564 * . there is enough space below the subdiagonal 00565 * . to fit an NS-by-NS scratch array.) ==== 00566 * 00567 IF( KBOT-KS+1.LE.NS / 2 ) THEN 00568 KS = KBOT - NS + 1 00569 KT = N - NS + 1 00570 CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH, 00571 $ H( KT, 1 ), LDH ) 00572 IF( NS.GT.NMIN ) THEN 00573 CALL ZLAQR4( .false., .false., NS, 1, NS, 00574 $ H( KT, 1 ), LDH, W( KS ), 1, 1, 00575 $ ZDUM, 1, WORK, LWORK, INF ) 00576 ELSE 00577 CALL ZLAHQR( .false., .false., NS, 1, NS, 00578 $ H( KT, 1 ), LDH, W( KS ), 1, 1, 00579 $ ZDUM, 1, INF ) 00580 END IF 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 ZLAQR0 ==== 00702 * 00703 END