![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief \b CLAQR0 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download CLAQR0 + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claqr0.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claqr0.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claqr0.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE CLAQR0( 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 *> \par Purpose: 00034 * ============= 00035 *> 00036 *> \verbatim 00037 *> 00038 *> CLAQR0 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 *> It is assumed that H is already upper triangular in rows 00081 *> and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1, 00082 *> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a 00083 *> previous call to CGEBAL, and then passed to CGEHRD when the 00084 *> matrix output by CGEBAL is reduced to Hessenberg form. 00085 *> Otherwise, ILO and IHI should be set to 1 and N, 00086 *> respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. 00087 *> If N = 0, then ILO = 1 and IHI = 0. 00088 *> \endverbatim 00089 *> 00090 *> \param[in,out] H 00091 *> \verbatim 00092 *> H is COMPLEX array, dimension (LDH,N) 00093 *> On entry, the upper Hessenberg matrix H. 00094 *> On exit, if INFO = 0 and WANTT is .TRUE., then H 00095 *> contains the upper triangular matrix T from the Schur 00096 *> decomposition (the Schur form). If INFO = 0 and WANT is 00097 *> .FALSE., then the contents of H are unspecified on exit. 00098 *> (The output value of H when INFO.GT.0 is given under the 00099 *> description of INFO below.) 00100 *> 00101 *> This subroutine may explicitly set H(i,j) = 0 for i.GT.j and 00102 *> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. 00103 *> \endverbatim 00104 *> 00105 *> \param[in] LDH 00106 *> \verbatim 00107 *> LDH is INTEGER 00108 *> The leading dimension of the array H. LDH .GE. max(1,N). 00109 *> \endverbatim 00110 *> 00111 *> \param[out] W 00112 *> \verbatim 00113 *> W is COMPLEX array, dimension (N) 00114 *> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored 00115 *> in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are 00116 *> stored in the same order as on the diagonal of the Schur 00117 *> form returned in H, with W(i) = H(i,i). 00118 *> \endverbatim 00119 *> 00120 *> \param[in] ILOZ 00121 *> \verbatim 00122 *> ILOZ is INTEGER 00123 *> \endverbatim 00124 *> 00125 *> \param[in] IHIZ 00126 *> \verbatim 00127 *> IHIZ is INTEGER 00128 *> Specify the rows of Z to which transformations must be 00129 *> applied if WANTZ is .TRUE.. 00130 *> 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. 00131 *> \endverbatim 00132 *> 00133 *> \param[in,out] Z 00134 *> \verbatim 00135 *> Z is COMPLEX array, dimension (LDZ,IHI) 00136 *> If WANTZ is .FALSE., then Z is not referenced. 00137 *> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is 00138 *> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the 00139 *> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). 00140 *> (The output value of Z when INFO.GT.0 is given under 00141 *> the description of INFO below.) 00142 *> \endverbatim 00143 *> 00144 *> \param[in] LDZ 00145 *> \verbatim 00146 *> LDZ is INTEGER 00147 *> The leading dimension of the array Z. if WANTZ is .TRUE. 00148 *> then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. 00149 *> \endverbatim 00150 *> 00151 *> \param[out] WORK 00152 *> \verbatim 00153 *> WORK is COMPLEX array, dimension LWORK 00154 *> On exit, if LWORK = -1, WORK(1) returns an estimate of 00155 *> the optimal value for LWORK. 00156 *> \endverbatim 00157 *> 00158 *> \param[in] LWORK 00159 *> \verbatim 00160 *> LWORK is INTEGER 00161 *> The dimension of the array WORK. LWORK .GE. max(1,N) 00162 *> is sufficient, but LWORK typically as large as 6*N may 00163 *> be required for optimal performance. A workspace query 00164 *> to determine the optimal workspace size is recommended. 00165 *> 00166 *> If LWORK = -1, then CLAQR0 does a workspace query. 00167 *> In this case, CLAQR0 checks the input parameters and 00168 *> estimates the optimal workspace size for the given 00169 *> values of N, ILO and IHI. The estimate is returned 00170 *> in WORK(1). No error message related to LWORK is 00171 *> issued by XERBLA. Neither H nor Z are accessed. 00172 *> \endverbatim 00173 *> 00174 *> \param[out] INFO 00175 *> \verbatim 00176 *> INFO is INTEGER 00177 *> = 0: successful exit 00178 *> .GT. 0: if INFO = i, CLAQR0 failed to compute all of 00179 *> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR 00180 *> and WI contain those eigenvalues which have been 00181 *> successfully computed. (Failures are rare.) 00182 *> 00183 *> If INFO .GT. 0 and WANT is .FALSE., then on exit, 00184 *> the remaining unconverged eigenvalues are the eigen- 00185 *> values of the upper Hessenberg matrix rows and 00186 *> columns ILO through INFO of the final, output 00187 *> value of H. 00188 *> 00189 *> If INFO .GT. 0 and WANTT is .TRUE., then on exit 00190 *> 00191 *> (*) (initial value of H)*U = U*(final value of H) 00192 *> 00193 *> where U is a unitary matrix. The final 00194 *> value of H is upper Hessenberg and triangular in 00195 *> rows and columns INFO+1 through IHI. 00196 *> 00197 *> If INFO .GT. 0 and WANTZ is .TRUE., then on exit 00198 *> 00199 *> (final value of Z(ILO:IHI,ILOZ:IHIZ) 00200 *> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U 00201 *> 00202 *> where U is the unitary matrix in (*) (regard- 00203 *> less of the value of WANTT.) 00204 *> 00205 *> If INFO .GT. 0 and WANTZ is .FALSE., then Z is not 00206 *> accessed. 00207 *> \endverbatim 00208 * 00209 * Authors: 00210 * ======== 00211 * 00212 *> \author Univ. of Tennessee 00213 *> \author Univ. of California Berkeley 00214 *> \author Univ. of Colorado Denver 00215 *> \author NAG Ltd. 00216 * 00217 *> \date November 2011 00218 * 00219 *> \ingroup complexOTHERauxiliary 00220 * 00221 *> \par Contributors: 00222 * ================== 00223 *> 00224 *> Karen Braman and Ralph Byers, Department of Mathematics, 00225 *> University of Kansas, USA 00226 * 00227 *> \par References: 00228 * ================ 00229 *> 00230 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 00231 *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 00232 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages 00233 *> 929--947, 2002. 00234 *> \n 00235 *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 00236 *> Algorithm Part II: Aggressive Early Deflation, SIAM Journal 00237 *> of Matrix Analysis, volume 23, pages 948--973, 2002. 00238 *> 00239 * ===================================================================== 00240 SUBROUTINE CLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 00241 $ IHIZ, Z, LDZ, WORK, LWORK, INFO ) 00242 * 00243 * -- LAPACK auxiliary routine (version 3.4.0) -- 00244 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00245 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00246 * November 2011 00247 * 00248 * .. Scalar Arguments .. 00249 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N 00250 LOGICAL WANTT, WANTZ 00251 * .. 00252 * .. Array Arguments .. 00253 COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) 00254 * .. 00255 * 00256 * ================================================================ 00257 * .. Parameters .. 00258 * 00259 * ==== Matrices of order NTINY or smaller must be processed by 00260 * . CLAHQR because of insufficient subdiagonal scratch space. 00261 * . (This is a hard limit.) ==== 00262 INTEGER NTINY 00263 PARAMETER ( NTINY = 11 ) 00264 * 00265 * ==== Exceptional deflation windows: try to cure rare 00266 * . slow convergence by varying the size of the 00267 * . deflation window after KEXNW iterations. ==== 00268 INTEGER KEXNW 00269 PARAMETER ( KEXNW = 5 ) 00270 * 00271 * ==== Exceptional shifts: try to cure rare slow convergence 00272 * . with ad-hoc exceptional shifts every KEXSH iterations. 00273 * . ==== 00274 INTEGER KEXSH 00275 PARAMETER ( KEXSH = 6 ) 00276 * 00277 * ==== The constant WILK1 is used to form the exceptional 00278 * . shifts. ==== 00279 REAL WILK1 00280 PARAMETER ( WILK1 = 0.75e0 ) 00281 COMPLEX ZERO, ONE 00282 PARAMETER ( ZERO = ( 0.0e0, 0.0e0 ), 00283 $ ONE = ( 1.0e0, 0.0e0 ) ) 00284 REAL TWO 00285 PARAMETER ( TWO = 2.0e0 ) 00286 * .. 00287 * .. Local Scalars .. 00288 COMPLEX AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2 00289 REAL S 00290 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, 00291 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, 00292 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS, 00293 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD 00294 LOGICAL SORTED 00295 CHARACTER JBCMPZ*2 00296 * .. 00297 * .. External Functions .. 00298 INTEGER ILAENV 00299 EXTERNAL ILAENV 00300 * .. 00301 * .. Local Arrays .. 00302 COMPLEX ZDUM( 1, 1 ) 00303 * .. 00304 * .. External Subroutines .. 00305 EXTERNAL CLACPY, CLAHQR, CLAQR3, CLAQR4, CLAQR5 00306 * .. 00307 * .. Intrinsic Functions .. 00308 INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, MIN, MOD, REAL, 00309 $ SQRT 00310 * .. 00311 * .. Statement Functions .. 00312 REAL CABS1 00313 * .. 00314 * .. Statement Function definitions .. 00315 CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) ) 00316 * .. 00317 * .. Executable Statements .. 00318 INFO = 0 00319 * 00320 * ==== Quick return for N = 0: nothing to do. ==== 00321 * 00322 IF( N.EQ.0 ) THEN 00323 WORK( 1 ) = ONE 00324 RETURN 00325 END IF 00326 * 00327 IF( N.LE.NTINY ) THEN 00328 * 00329 * ==== Tiny matrices must use CLAHQR. ==== 00330 * 00331 LWKOPT = 1 00332 IF( LWORK.NE.-1 ) 00333 $ CALL CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 00334 $ IHIZ, Z, LDZ, INFO ) 00335 ELSE 00336 * 00337 * ==== Use small bulge multi-shift QR with aggressive early 00338 * . deflation on larger-than-tiny matrices. ==== 00339 * 00340 * ==== Hope for the best. ==== 00341 * 00342 INFO = 0 00343 * 00344 * ==== Set up job flags for ILAENV. ==== 00345 * 00346 IF( WANTT ) THEN 00347 JBCMPZ( 1: 1 ) = 'S' 00348 ELSE 00349 JBCMPZ( 1: 1 ) = 'E' 00350 END IF 00351 IF( WANTZ ) THEN 00352 JBCMPZ( 2: 2 ) = 'V' 00353 ELSE 00354 JBCMPZ( 2: 2 ) = 'N' 00355 END IF 00356 * 00357 * ==== NWR = recommended deflation window size. At this 00358 * . point, N .GT. NTINY = 11, so there is enough 00359 * . subdiagonal workspace for NWR.GE.2 as required. 00360 * . (In fact, there is enough subdiagonal space for 00361 * . NWR.GE.3.) ==== 00362 * 00363 NWR = ILAENV( 13, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00364 NWR = MAX( 2, NWR ) 00365 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) 00366 * 00367 * ==== NSR = recommended number of simultaneous shifts. 00368 * . At this point N .GT. NTINY = 11, so there is at 00369 * . enough subdiagonal workspace for NSR to be even 00370 * . and greater than or equal to two as required. ==== 00371 * 00372 NSR = ILAENV( 15, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00373 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO ) 00374 NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) 00375 * 00376 * ==== Estimate optimal workspace ==== 00377 * 00378 * ==== Workspace query call to CLAQR3 ==== 00379 * 00380 CALL CLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, 00381 $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H, 00382 $ LDH, WORK, -1 ) 00383 * 00384 * ==== Optimal workspace = MAX(CLAQR5, CLAQR3) ==== 00385 * 00386 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) 00387 * 00388 * ==== Quick return in case of workspace query. ==== 00389 * 00390 IF( LWORK.EQ.-1 ) THEN 00391 WORK( 1 ) = CMPLX( LWKOPT, 0 ) 00392 RETURN 00393 END IF 00394 * 00395 * ==== CLAHQR/CLAQR0 crossover point ==== 00396 * 00397 NMIN = ILAENV( 12, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00398 NMIN = MAX( NTINY, NMIN ) 00399 * 00400 * ==== Nibble crossover point ==== 00401 * 00402 NIBBLE = ILAENV( 14, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00403 NIBBLE = MAX( 0, NIBBLE ) 00404 * 00405 * ==== Accumulate reflections during ttswp? Use block 00406 * . 2-by-2 structure during matrix-matrix multiply? ==== 00407 * 00408 KACC22 = ILAENV( 16, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 00409 KACC22 = MAX( 0, KACC22 ) 00410 KACC22 = MIN( 2, KACC22 ) 00411 * 00412 * ==== NWMAX = the largest possible deflation window for 00413 * . which there is sufficient workspace. ==== 00414 * 00415 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) 00416 NW = NWMAX 00417 * 00418 * ==== NSMAX = the Largest number of simultaneous shifts 00419 * . for which there is sufficient workspace. ==== 00420 * 00421 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 ) 00422 NSMAX = NSMAX - MOD( NSMAX, 2 ) 00423 * 00424 * ==== NDFL: an iteration count restarted at deflation. ==== 00425 * 00426 NDFL = 1 00427 * 00428 * ==== ITMAX = iteration limit ==== 00429 * 00430 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) 00431 * 00432 * ==== Last row and column in the active block ==== 00433 * 00434 KBOT = IHI 00435 * 00436 * ==== Main Loop ==== 00437 * 00438 DO 70 IT = 1, ITMAX 00439 * 00440 * ==== Done when KBOT falls below ILO ==== 00441 * 00442 IF( KBOT.LT.ILO ) 00443 $ GO TO 80 00444 * 00445 * ==== Locate active block ==== 00446 * 00447 DO 10 K = KBOT, ILO + 1, -1 00448 IF( H( K, K-1 ).EQ.ZERO ) 00449 $ GO TO 20 00450 10 CONTINUE 00451 K = ILO 00452 20 CONTINUE 00453 KTOP = K 00454 * 00455 * ==== Select deflation window size: 00456 * . Typical Case: 00457 * . If possible and advisable, nibble the entire 00458 * . active block. If not, use size MIN(NWR,NWMAX) 00459 * . or MIN(NWR+1,NWMAX) depending upon which has 00460 * . the smaller corresponding subdiagonal entry 00461 * . (a heuristic). 00462 * . 00463 * . Exceptional Case: 00464 * . If there have been no deflations in KEXNW or 00465 * . more iterations, then vary the deflation window 00466 * . size. At first, because, larger windows are, 00467 * . in general, more powerful than smaller ones, 00468 * . rapidly increase the window to the maximum possible. 00469 * . Then, gradually reduce the window size. ==== 00470 * 00471 NH = KBOT - KTOP + 1 00472 NWUPBD = MIN( NH, NWMAX ) 00473 IF( NDFL.LT.KEXNW ) THEN 00474 NW = MIN( NWUPBD, NWR ) 00475 ELSE 00476 NW = MIN( NWUPBD, 2*NW ) 00477 END IF 00478 IF( NW.LT.NWMAX ) THEN 00479 IF( NW.GE.NH-1 ) THEN 00480 NW = NH 00481 ELSE 00482 KWTOP = KBOT - NW + 1 00483 IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT. 00484 $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 00485 END IF 00486 END IF 00487 IF( NDFL.LT.KEXNW ) THEN 00488 NDEC = -1 00489 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN 00490 NDEC = NDEC + 1 00491 IF( NW-NDEC.LT.2 ) 00492 $ NDEC = 0 00493 NW = NW - NDEC 00494 END IF 00495 * 00496 * ==== Aggressive early deflation: 00497 * . split workspace under the subdiagonal into 00498 * . - an nw-by-nw work array V in the lower 00499 * . left-hand-corner, 00500 * . - an NW-by-at-least-NW-but-more-is-better 00501 * . (NW-by-NHO) horizontal work array along 00502 * . the bottom edge, 00503 * . - an at-least-NW-but-more-is-better (NHV-by-NW) 00504 * . vertical work array along the left-hand-edge. 00505 * . ==== 00506 * 00507 KV = N - NW + 1 00508 KT = NW + 1 00509 NHO = ( N-NW-1 ) - KT + 1 00510 KWV = NW + 2 00511 NVE = ( N-NW ) - KWV + 1 00512 * 00513 * ==== Aggressive early deflation ==== 00514 * 00515 CALL CLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, 00516 $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO, 00517 $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK, 00518 $ LWORK ) 00519 * 00520 * ==== Adjust KBOT accounting for new deflations. ==== 00521 * 00522 KBOT = KBOT - LD 00523 * 00524 * ==== KS points to the shifts. ==== 00525 * 00526 KS = KBOT - LS + 1 00527 * 00528 * ==== Skip an expensive QR sweep if there is a (partly 00529 * . heuristic) reason to expect that many eigenvalues 00530 * . will deflate without it. Here, the QR sweep is 00531 * . skipped if many eigenvalues have just been deflated 00532 * . or if the remaining active block is small. 00533 * 00534 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- 00535 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN 00536 * 00537 * ==== NS = nominal number of simultaneous shifts. 00538 * . This may be lowered (slightly) if CLAQR3 00539 * . did not provide that many shifts. ==== 00540 * 00541 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) 00542 NS = NS - MOD( NS, 2 ) 00543 * 00544 * ==== If there have been no deflations 00545 * . in a multiple of KEXSH iterations, 00546 * . then try exceptional shifts. 00547 * . Otherwise use shifts provided by 00548 * . CLAQR3 above or from the eigenvalues 00549 * . of a trailing principal submatrix. ==== 00550 * 00551 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN 00552 KS = KBOT - NS + 1 00553 DO 30 I = KBOT, KS + 1, -2 00554 W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) ) 00555 W( I-1 ) = W( I ) 00556 30 CONTINUE 00557 ELSE 00558 * 00559 * ==== Got NS/2 or fewer shifts? Use CLAQR4 or 00560 * . CLAHQR on a trailing principal submatrix to 00561 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, 00562 * . there is enough space below the subdiagonal 00563 * . to fit an NS-by-NS scratch array.) ==== 00564 * 00565 IF( KBOT-KS+1.LE.NS / 2 ) THEN 00566 KS = KBOT - NS + 1 00567 KT = N - NS + 1 00568 CALL CLACPY( 'A', NS, NS, H( KS, KS ), LDH, 00569 $ H( KT, 1 ), LDH ) 00570 IF( NS.GT.NMIN ) THEN 00571 CALL CLAQR4( .false., .false., NS, 1, NS, 00572 $ H( KT, 1 ), LDH, W( KS ), 1, 1, 00573 $ ZDUM, 1, WORK, LWORK, INF ) 00574 ELSE 00575 CALL CLAHQR( .false., .false., NS, 1, NS, 00576 $ H( KT, 1 ), LDH, W( KS ), 1, 1, 00577 $ ZDUM, 1, INF ) 00578 END IF 00579 KS = KS + INF 00580 * 00581 * ==== In case of a rare QR failure use 00582 * . eigenvalues of the trailing 2-by-2 00583 * . principal submatrix. Scale to avoid 00584 * . overflows, underflows and subnormals. 00585 * . (The scale factor S can not be zero, 00586 * . because H(KBOT,KBOT-1) is nonzero.) ==== 00587 * 00588 IF( KS.GE.KBOT ) THEN 00589 S = CABS1( H( KBOT-1, KBOT-1 ) ) + 00590 $ CABS1( H( KBOT, KBOT-1 ) ) + 00591 $ CABS1( H( KBOT-1, KBOT ) ) + 00592 $ CABS1( H( KBOT, KBOT ) ) 00593 AA = H( KBOT-1, KBOT-1 ) / S 00594 CC = H( KBOT, KBOT-1 ) / S 00595 BB = H( KBOT-1, KBOT ) / S 00596 DD = H( KBOT, KBOT ) / S 00597 TR2 = ( AA+DD ) / TWO 00598 DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC 00599 RTDISC = SQRT( -DET ) 00600 W( KBOT-1 ) = ( TR2+RTDISC )*S 00601 W( KBOT ) = ( TR2-RTDISC )*S 00602 * 00603 KS = KBOT - 1 00604 END IF 00605 END IF 00606 * 00607 IF( KBOT-KS+1.GT.NS ) THEN 00608 * 00609 * ==== Sort the shifts (Helps a little) ==== 00610 * 00611 SORTED = .false. 00612 DO 50 K = KBOT, KS + 1, -1 00613 IF( SORTED ) 00614 $ GO TO 60 00615 SORTED = .true. 00616 DO 40 I = KS, K - 1 00617 IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) ) 00618 $ THEN 00619 SORTED = .false. 00620 SWAP = W( I ) 00621 W( I ) = W( I+1 ) 00622 W( I+1 ) = SWAP 00623 END IF 00624 40 CONTINUE 00625 50 CONTINUE 00626 60 CONTINUE 00627 END IF 00628 END IF 00629 * 00630 * ==== If there are only two shifts, then use 00631 * . only one. ==== 00632 * 00633 IF( KBOT-KS+1.EQ.2 ) THEN 00634 IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT. 00635 $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN 00636 W( KBOT-1 ) = W( KBOT ) 00637 ELSE 00638 W( KBOT ) = W( KBOT-1 ) 00639 END IF 00640 END IF 00641 * 00642 * ==== Use up to NS of the the smallest magnatiude 00643 * . shifts. If there aren't NS shifts available, 00644 * . then use them all, possibly dropping one to 00645 * . make the number of shifts even. ==== 00646 * 00647 NS = MIN( NS, KBOT-KS+1 ) 00648 NS = NS - MOD( NS, 2 ) 00649 KS = KBOT - NS + 1 00650 * 00651 * ==== Small-bulge multi-shift QR sweep: 00652 * . split workspace under the subdiagonal into 00653 * . - a KDU-by-KDU work array U in the lower 00654 * . left-hand-corner, 00655 * . - a KDU-by-at-least-KDU-but-more-is-better 00656 * . (KDU-by-NHo) horizontal work array WH along 00657 * . the bottom edge, 00658 * . - and an at-least-KDU-but-more-is-better-by-KDU 00659 * . (NVE-by-KDU) vertical work WV arrow along 00660 * . the left-hand-edge. ==== 00661 * 00662 KDU = 3*NS - 3 00663 KU = N - KDU + 1 00664 KWH = KDU + 1 00665 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 00666 KWV = KDU + 4 00667 NVE = N - KDU - KWV + 1 00668 * 00669 * ==== Small-bulge multi-shift QR sweep ==== 00670 * 00671 CALL CLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, 00672 $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK, 00673 $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH, 00674 $ NHO, H( KU, KWH ), LDH ) 00675 END IF 00676 * 00677 * ==== Note progress (or the lack of it). ==== 00678 * 00679 IF( LD.GT.0 ) THEN 00680 NDFL = 1 00681 ELSE 00682 NDFL = NDFL + 1 00683 END IF 00684 * 00685 * ==== End of main loop ==== 00686 70 CONTINUE 00687 * 00688 * ==== Iteration limit exceeded. Set INFO to show where 00689 * . the problem occurred and exit. ==== 00690 * 00691 INFO = KBOT 00692 80 CONTINUE 00693 END IF 00694 * 00695 * ==== Return the optimal value of LWORK. ==== 00696 * 00697 WORK( 1 ) = CMPLX( LWKOPT, 0 ) 00698 * 00699 * ==== End of CLAQR0 ==== 00700 * 00701 END