1*> \brief \b CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download CLAQR4 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/claqr4.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/claqr4.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/claqr4.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 22* IHIZ, Z, LDZ, WORK, LWORK, INFO ) 23* 24* .. Scalar Arguments .. 25* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N 26* LOGICAL WANTT, WANTZ 27* .. 28* .. Array Arguments .. 29* COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) 30* .. 31* 32* 33* 34*> \par Purpose: 35* ============= 36*> 37*> \verbatim 38*> 39*> CLAQR4 implements one level of recursion for CLAQR0. 40*> It is a complete implementation of the small bulge multi-shift 41*> QR algorithm. It may be called by CLAQR0 and, for large enough 42*> deflation window size, it may be called by CLAQR3. This 43*> subroutine is identical to CLAQR0 except that it calls CLAQR2 44*> instead of CLAQR3. 45*> 46*> CLAQR4 computes the eigenvalues of a Hessenberg matrix H 47*> and, optionally, the matrices T and Z from the Schur decomposition 48*> H = Z T Z**H, where T is an upper triangular matrix (the 49*> Schur form), and Z is the unitary matrix of Schur vectors. 50*> 51*> Optionally Z may be postmultiplied into an input unitary 52*> matrix Q so that this routine can give the Schur factorization 53*> of a matrix A which has been reduced to the Hessenberg form H 54*> by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. 55*> \endverbatim 56* 57* Arguments: 58* ========== 59* 60*> \param[in] WANTT 61*> \verbatim 62*> WANTT is LOGICAL 63*> = .TRUE. : the full Schur form T is required; 64*> = .FALSE.: only eigenvalues are required. 65*> \endverbatim 66*> 67*> \param[in] WANTZ 68*> \verbatim 69*> WANTZ is LOGICAL 70*> = .TRUE. : the matrix of Schur vectors Z is required; 71*> = .FALSE.: Schur vectors are not required. 72*> \endverbatim 73*> 74*> \param[in] N 75*> \verbatim 76*> N is INTEGER 77*> The order of the matrix H. N >= 0. 78*> \endverbatim 79*> 80*> \param[in] ILO 81*> \verbatim 82*> ILO is INTEGER 83*> \endverbatim 84*> 85*> \param[in] IHI 86*> \verbatim 87*> IHI is INTEGER 88*> It is assumed that H is already upper triangular in rows 89*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, 90*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a 91*> previous call to CGEBAL, and then passed to CGEHRD when the 92*> matrix output by CGEBAL is reduced to Hessenberg form. 93*> Otherwise, ILO and IHI should be set to 1 and N, 94*> respectively. If N > 0, then 1 <= ILO <= IHI <= N. 95*> If N = 0, then ILO = 1 and IHI = 0. 96*> \endverbatim 97*> 98*> \param[in,out] H 99*> \verbatim 100*> H is COMPLEX array, dimension (LDH,N) 101*> On entry, the upper Hessenberg matrix H. 102*> On exit, if INFO = 0 and WANTT is .TRUE., then H 103*> contains the upper triangular matrix T from the Schur 104*> decomposition (the Schur form). If INFO = 0 and WANT is 105*> .FALSE., then the contents of H are unspecified on exit. 106*> (The output value of H when INFO > 0 is given under the 107*> description of INFO below.) 108*> 109*> This subroutine may explicitly set H(i,j) = 0 for i > j and 110*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. 111*> \endverbatim 112*> 113*> \param[in] LDH 114*> \verbatim 115*> LDH is INTEGER 116*> The leading dimension of the array H. LDH >= max(1,N). 117*> \endverbatim 118*> 119*> \param[out] W 120*> \verbatim 121*> W is COMPLEX array, dimension (N) 122*> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored 123*> in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are 124*> stored in the same order as on the diagonal of the Schur 125*> form returned in H, with W(i) = H(i,i). 126*> \endverbatim 127*> 128*> \param[in] ILOZ 129*> \verbatim 130*> ILOZ is INTEGER 131*> \endverbatim 132*> 133*> \param[in] IHIZ 134*> \verbatim 135*> IHIZ is INTEGER 136*> Specify the rows of Z to which transformations must be 137*> applied if WANTZ is .TRUE.. 138*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. 139*> \endverbatim 140*> 141*> \param[in,out] Z 142*> \verbatim 143*> Z is COMPLEX array, dimension (LDZ,IHI) 144*> If WANTZ is .FALSE., then Z is not referenced. 145*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is 146*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the 147*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). 148*> (The output value of Z when INFO > 0 is given under 149*> the description of INFO below.) 150*> \endverbatim 151*> 152*> \param[in] LDZ 153*> \verbatim 154*> LDZ is INTEGER 155*> The leading dimension of the array Z. if WANTZ is .TRUE. 156*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. 157*> \endverbatim 158*> 159*> \param[out] WORK 160*> \verbatim 161*> WORK is COMPLEX array, dimension LWORK 162*> On exit, if LWORK = -1, WORK(1) returns an estimate of 163*> the optimal value for LWORK. 164*> \endverbatim 165*> 166*> \param[in] LWORK 167*> \verbatim 168*> LWORK is INTEGER 169*> The dimension of the array WORK. LWORK >= max(1,N) 170*> is sufficient, but LWORK typically as large as 6*N may 171*> be required for optimal performance. A workspace query 172*> to determine the optimal workspace size is recommended. 173*> 174*> If LWORK = -1, then CLAQR4 does a workspace query. 175*> In this case, CLAQR4 checks the input parameters and 176*> estimates the optimal workspace size for the given 177*> values of N, ILO and IHI. The estimate is returned 178*> in WORK(1). No error message related to LWORK is 179*> issued by XERBLA. Neither H nor Z are accessed. 180*> \endverbatim 181*> 182*> \param[out] INFO 183*> \verbatim 184*> INFO is INTEGER 185*> = 0: successful exit 186*> > 0: if INFO = i, CLAQR4 failed to compute all of 187*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR 188*> and WI contain those eigenvalues which have been 189*> successfully computed. (Failures are rare.) 190*> 191*> If INFO > 0 and WANT is .FALSE., then on exit, 192*> the remaining unconverged eigenvalues are the eigen- 193*> values of the upper Hessenberg matrix rows and 194*> columns ILO through INFO of the final, output 195*> value of H. 196*> 197*> If INFO > 0 and WANTT is .TRUE., then on exit 198*> 199*> (*) (initial value of H)*U = U*(final value of H) 200*> 201*> where U is a unitary matrix. The final 202*> value of H is upper Hessenberg and triangular in 203*> rows and columns INFO+1 through IHI. 204*> 205*> If INFO > 0 and WANTZ is .TRUE., then on exit 206*> 207*> (final value of Z(ILO:IHI,ILOZ:IHIZ) 208*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U 209*> 210*> where U is the unitary matrix in (*) (regard- 211*> less of the value of WANTT.) 212*> 213*> If INFO > 0 and WANTZ is .FALSE., then Z is not 214*> accessed. 215*> \endverbatim 216* 217* Authors: 218* ======== 219* 220*> \author Univ. of Tennessee 221*> \author Univ. of California Berkeley 222*> \author Univ. of Colorado Denver 223*> \author NAG Ltd. 224* 225*> \ingroup complexOTHERauxiliary 226* 227*> \par Contributors: 228* ================== 229*> 230*> Karen Braman and Ralph Byers, Department of Mathematics, 231*> University of Kansas, USA 232* 233*> \par References: 234* ================ 235*> 236*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 237*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 238*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages 239*> 929--947, 2002. 240*> \n 241*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 242*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal 243*> of Matrix Analysis, volume 23, pages 948--973, 2002. 244*> 245* ===================================================================== 246 SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 247 $ IHIZ, Z, LDZ, WORK, LWORK, INFO ) 248* 249* -- LAPACK auxiliary routine -- 250* -- LAPACK is a software package provided by Univ. of Tennessee, -- 251* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 252* 253* .. Scalar Arguments .. 254 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N 255 LOGICAL WANTT, WANTZ 256* .. 257* .. Array Arguments .. 258 COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) 259* .. 260* 261* 262* ================================================================ 263* 264* .. Parameters .. 265* 266* ==== Matrices of order NTINY or smaller must be processed by 267* . CLAHQR because of insufficient subdiagonal scratch space. 268* . (This is a hard limit.) ==== 269 INTEGER NTINY 270 PARAMETER ( NTINY = 15 ) 271* 272* ==== Exceptional deflation windows: try to cure rare 273* . slow convergence by varying the size of the 274* . deflation window after KEXNW iterations. ==== 275 INTEGER KEXNW 276 PARAMETER ( KEXNW = 5 ) 277* 278* ==== Exceptional shifts: try to cure rare slow convergence 279* . with ad-hoc exceptional shifts every KEXSH iterations. 280* . ==== 281 INTEGER KEXSH 282 PARAMETER ( KEXSH = 6 ) 283* 284* ==== The constant WILK1 is used to form the exceptional 285* . shifts. ==== 286 REAL WILK1 287 PARAMETER ( WILK1 = 0.75e0 ) 288 COMPLEX ZERO, ONE 289 PARAMETER ( ZERO = ( 0.0e0, 0.0e0 ), 290 $ ONE = ( 1.0e0, 0.0e0 ) ) 291 REAL TWO 292 PARAMETER ( TWO = 2.0e0 ) 293* .. 294* .. Local Scalars .. 295 COMPLEX AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2 296 REAL S 297 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, 298 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, 299 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS, 300 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD 301 LOGICAL SORTED 302 CHARACTER JBCMPZ*2 303* .. 304* .. External Functions .. 305 INTEGER ILAENV 306 EXTERNAL ILAENV 307* .. 308* .. Local Arrays .. 309 COMPLEX ZDUM( 1, 1 ) 310* .. 311* .. External Subroutines .. 312 EXTERNAL CLACPY, CLAHQR, CLAQR2, CLAQR5 313* .. 314* .. Intrinsic Functions .. 315 INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, MIN, MOD, REAL, 316 $ SQRT 317* .. 318* .. Statement Functions .. 319 REAL CABS1 320* .. 321* .. Statement Function definitions .. 322 CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) ) 323* .. 324* .. Executable Statements .. 325 INFO = 0 326* 327* ==== Quick return for N = 0: nothing to do. ==== 328* 329 IF( N.EQ.0 ) THEN 330 WORK( 1 ) = ONE 331 RETURN 332 END IF 333* 334 IF( N.LE.NTINY ) THEN 335* 336* ==== Tiny matrices must use CLAHQR. ==== 337* 338 LWKOPT = 1 339 IF( LWORK.NE.-1 ) 340 $ CALL CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 341 $ IHIZ, Z, LDZ, INFO ) 342 ELSE 343* 344* ==== Use small bulge multi-shift QR with aggressive early 345* . deflation on larger-than-tiny matrices. ==== 346* 347* ==== Hope for the best. ==== 348* 349 INFO = 0 350* 351* ==== Set up job flags for ILAENV. ==== 352* 353 IF( WANTT ) THEN 354 JBCMPZ( 1: 1 ) = 'S' 355 ELSE 356 JBCMPZ( 1: 1 ) = 'E' 357 END IF 358 IF( WANTZ ) THEN 359 JBCMPZ( 2: 2 ) = 'V' 360 ELSE 361 JBCMPZ( 2: 2 ) = 'N' 362 END IF 363* 364* ==== NWR = recommended deflation window size. At this 365* . point, N .GT. NTINY = 15, so there is enough 366* . subdiagonal workspace for NWR.GE.2 as required. 367* . (In fact, there is enough subdiagonal space for 368* . NWR.GE.4.) ==== 369* 370 NWR = ILAENV( 13, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) 371 NWR = MAX( 2, NWR ) 372 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) 373* 374* ==== NSR = recommended number of simultaneous shifts. 375* . At this point N .GT. NTINY = 15, so there is at 376* . enough subdiagonal workspace for NSR to be even 377* . and greater than or equal to two as required. ==== 378* 379 NSR = ILAENV( 15, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) 380 NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO ) 381 NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) 382* 383* ==== Estimate optimal workspace ==== 384* 385* ==== Workspace query call to CLAQR2 ==== 386* 387 CALL CLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, 388 $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H, 389 $ LDH, WORK, -1 ) 390* 391* ==== Optimal workspace = MAX(CLAQR5, CLAQR2) ==== 392* 393 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) 394* 395* ==== Quick return in case of workspace query. ==== 396* 397 IF( LWORK.EQ.-1 ) THEN 398 WORK( 1 ) = CMPLX( LWKOPT, 0 ) 399 RETURN 400 END IF 401* 402* ==== CLAHQR/CLAQR0 crossover point ==== 403* 404 NMIN = ILAENV( 12, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) 405 NMIN = MAX( NTINY, NMIN ) 406* 407* ==== Nibble crossover point ==== 408* 409 NIBBLE = ILAENV( 14, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) 410 NIBBLE = MAX( 0, NIBBLE ) 411* 412* ==== Accumulate reflections during ttswp? Use block 413* . 2-by-2 structure during matrix-matrix multiply? ==== 414* 415 KACC22 = ILAENV( 16, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) 416 KACC22 = MAX( 0, KACC22 ) 417 KACC22 = MIN( 2, KACC22 ) 418* 419* ==== NWMAX = the largest possible deflation window for 420* . which there is sufficient workspace. ==== 421* 422 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) 423 NW = NWMAX 424* 425* ==== NSMAX = the Largest number of simultaneous shifts 426* . for which there is sufficient workspace. ==== 427* 428 NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 ) 429 NSMAX = NSMAX - MOD( NSMAX, 2 ) 430* 431* ==== NDFL: an iteration count restarted at deflation. ==== 432* 433 NDFL = 1 434* 435* ==== ITMAX = iteration limit ==== 436* 437 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) 438* 439* ==== Last row and column in the active block ==== 440* 441 KBOT = IHI 442* 443* ==== Main Loop ==== 444* 445 DO 70 IT = 1, ITMAX 446* 447* ==== Done when KBOT falls below ILO ==== 448* 449 IF( KBOT.LT.ILO ) 450 $ GO TO 80 451* 452* ==== Locate active block ==== 453* 454 DO 10 K = KBOT, ILO + 1, -1 455 IF( H( K, K-1 ).EQ.ZERO ) 456 $ GO TO 20 457 10 CONTINUE 458 K = ILO 459 20 CONTINUE 460 KTOP = K 461* 462* ==== Select deflation window size: 463* . Typical Case: 464* . If possible and advisable, nibble the entire 465* . active block. If not, use size MIN(NWR,NWMAX) 466* . or MIN(NWR+1,NWMAX) depending upon which has 467* . the smaller corresponding subdiagonal entry 468* . (a heuristic). 469* . 470* . Exceptional Case: 471* . If there have been no deflations in KEXNW or 472* . more iterations, then vary the deflation window 473* . size. At first, because, larger windows are, 474* . in general, more powerful than smaller ones, 475* . rapidly increase the window to the maximum possible. 476* . Then, gradually reduce the window size. ==== 477* 478 NH = KBOT - KTOP + 1 479 NWUPBD = MIN( NH, NWMAX ) 480 IF( NDFL.LT.KEXNW ) THEN 481 NW = MIN( NWUPBD, NWR ) 482 ELSE 483 NW = MIN( NWUPBD, 2*NW ) 484 END IF 485 IF( NW.LT.NWMAX ) THEN 486 IF( NW.GE.NH-1 ) THEN 487 NW = NH 488 ELSE 489 KWTOP = KBOT - NW + 1 490 IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT. 491 $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 492 END IF 493 END IF 494 IF( NDFL.LT.KEXNW ) THEN 495 NDEC = -1 496 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN 497 NDEC = NDEC + 1 498 IF( NW-NDEC.LT.2 ) 499 $ NDEC = 0 500 NW = NW - NDEC 501 END IF 502* 503* ==== Aggressive early deflation: 504* . split workspace under the subdiagonal into 505* . - an nw-by-nw work array V in the lower 506* . left-hand-corner, 507* . - an NW-by-at-least-NW-but-more-is-better 508* . (NW-by-NHO) horizontal work array along 509* . the bottom edge, 510* . - an at-least-NW-but-more-is-better (NHV-by-NW) 511* . vertical work array along the left-hand-edge. 512* . ==== 513* 514 KV = N - NW + 1 515 KT = NW + 1 516 NHO = ( N-NW-1 ) - KT + 1 517 KWV = NW + 2 518 NVE = ( N-NW ) - KWV + 1 519* 520* ==== Aggressive early deflation ==== 521* 522 CALL CLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, 523 $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO, 524 $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK, 525 $ LWORK ) 526* 527* ==== Adjust KBOT accounting for new deflations. ==== 528* 529 KBOT = KBOT - LD 530* 531* ==== KS points to the shifts. ==== 532* 533 KS = KBOT - LS + 1 534* 535* ==== Skip an expensive QR sweep if there is a (partly 536* . heuristic) reason to expect that many eigenvalues 537* . will deflate without it. Here, the QR sweep is 538* . skipped if many eigenvalues have just been deflated 539* . or if the remaining active block is small. 540* 541 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- 542 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN 543* 544* ==== NS = nominal number of simultaneous shifts. 545* . This may be lowered (slightly) if CLAQR2 546* . did not provide that many shifts. ==== 547* 548 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) 549 NS = NS - MOD( NS, 2 ) 550* 551* ==== If there have been no deflations 552* . in a multiple of KEXSH iterations, 553* . then try exceptional shifts. 554* . Otherwise use shifts provided by 555* . CLAQR2 above or from the eigenvalues 556* . of a trailing principal submatrix. ==== 557* 558 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN 559 KS = KBOT - NS + 1 560 DO 30 I = KBOT, KS + 1, -2 561 W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) ) 562 W( I-1 ) = W( I ) 563 30 CONTINUE 564 ELSE 565* 566* ==== Got NS/2 or fewer shifts? Use CLAHQR 567* . on a trailing principal submatrix to 568* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6, 569* . there is enough space below the subdiagonal 570* . to fit an NS-by-NS scratch array.) ==== 571* 572 IF( KBOT-KS+1.LE.NS / 2 ) THEN 573 KS = KBOT - NS + 1 574 KT = N - NS + 1 575 CALL CLACPY( 'A', NS, NS, H( KS, KS ), LDH, 576 $ H( KT, 1 ), LDH ) 577 CALL CLAHQR( .false., .false., NS, 1, NS, 578 $ H( KT, 1 ), LDH, W( KS ), 1, 1, ZDUM, 579 $ 1, INF ) 580 KS = KS + INF 581* 582* ==== In case of a rare QR failure use 583* . eigenvalues of the trailing 2-by-2 584* . principal submatrix. Scale to avoid 585* . overflows, underflows and subnormals. 586* . (The scale factor S can not be zero, 587* . because H(KBOT,KBOT-1) is nonzero.) ==== 588* 589 IF( KS.GE.KBOT ) THEN 590 S = CABS1( H( KBOT-1, KBOT-1 ) ) + 591 $ CABS1( H( KBOT, KBOT-1 ) ) + 592 $ CABS1( H( KBOT-1, KBOT ) ) + 593 $ CABS1( H( KBOT, KBOT ) ) 594 AA = H( KBOT-1, KBOT-1 ) / S 595 CC = H( KBOT, KBOT-1 ) / S 596 BB = H( KBOT-1, KBOT ) / S 597 DD = H( KBOT, KBOT ) / S 598 TR2 = ( AA+DD ) / TWO 599 DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC 600 RTDISC = SQRT( -DET ) 601 W( KBOT-1 ) = ( TR2+RTDISC )*S 602 W( KBOT ) = ( TR2-RTDISC )*S 603* 604 KS = KBOT - 1 605 END IF 606 END IF 607* 608 IF( KBOT-KS+1.GT.NS ) THEN 609* 610* ==== Sort the shifts (Helps a little) ==== 611* 612 SORTED = .false. 613 DO 50 K = KBOT, KS + 1, -1 614 IF( SORTED ) 615 $ GO TO 60 616 SORTED = .true. 617 DO 40 I = KS, K - 1 618 IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) ) 619 $ THEN 620 SORTED = .false. 621 SWAP = W( I ) 622 W( I ) = W( I+1 ) 623 W( I+1 ) = SWAP 624 END IF 625 40 CONTINUE 626 50 CONTINUE 627 60 CONTINUE 628 END IF 629 END IF 630* 631* ==== If there are only two shifts, then use 632* . only one. ==== 633* 634 IF( KBOT-KS+1.EQ.2 ) THEN 635 IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT. 636 $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN 637 W( KBOT-1 ) = W( KBOT ) 638 ELSE 639 W( KBOT ) = W( KBOT-1 ) 640 END IF 641 END IF 642* 643* ==== Use up to NS of the the smallest magnitude 644* . shifts. If there aren't NS shifts available, 645* . then use them all, possibly dropping one to 646* . make the number of shifts even. ==== 647* 648 NS = MIN( NS, KBOT-KS+1 ) 649 NS = NS - MOD( NS, 2 ) 650 KS = KBOT - NS + 1 651* 652* ==== Small-bulge multi-shift QR sweep: 653* . split workspace under the subdiagonal into 654* . - a KDU-by-KDU work array U in the lower 655* . left-hand-corner, 656* . - a KDU-by-at-least-KDU-but-more-is-better 657* . (KDU-by-NHo) horizontal work array WH along 658* . the bottom edge, 659* . - and an at-least-KDU-but-more-is-better-by-KDU 660* . (NVE-by-KDU) vertical work WV arrow along 661* . the left-hand-edge. ==== 662* 663 KDU = 2*NS 664 KU = N - KDU + 1 665 KWH = KDU + 1 666 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 667 KWV = KDU + 4 668 NVE = N - KDU - KWV + 1 669* 670* ==== Small-bulge multi-shift QR sweep ==== 671* 672 CALL CLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, 673 $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK, 674 $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH, 675 $ NHO, H( KU, KWH ), LDH ) 676 END IF 677* 678* ==== Note progress (or the lack of it). ==== 679* 680 IF( LD.GT.0 ) THEN 681 NDFL = 1 682 ELSE 683 NDFL = NDFL + 1 684 END IF 685* 686* ==== End of main loop ==== 687 70 CONTINUE 688* 689* ==== Iteration limit exceeded. Set INFO to show where 690* . the problem occurred and exit. ==== 691* 692 INFO = KBOT 693 80 CONTINUE 694 END IF 695* 696* ==== Return the optimal value of LWORK. ==== 697* 698 WORK( 1 ) = CMPLX( LWKOPT, 0 ) 699* 700* ==== End of CLAQR4 ==== 701* 702 END 703