1*> \brief \b ZLAQR0 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 ZLAQR0 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr0.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr0.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr0.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZLAQR0( 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*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) 30* .. 31* 32* 33*> \par Purpose: 34* ============= 35*> 36*> \verbatim 37*> 38*> ZLAQR0 computes the eigenvalues of a Hessenberg matrix H 39*> and, optionally, the matrices T and Z from the Schur decomposition 40*> H = Z T Z**H, where T is an upper triangular matrix (the 41*> Schur form), and Z is the unitary matrix of Schur vectors. 42*> 43*> Optionally Z may be postmultiplied into an input unitary 44*> matrix Q so that this routine can give the Schur factorization 45*> of a matrix A which has been reduced to the Hessenberg form H 46*> by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. 47*> \endverbatim 48* 49* Arguments: 50* ========== 51* 52*> \param[in] WANTT 53*> \verbatim 54*> WANTT is LOGICAL 55*> = .TRUE. : the full Schur form T is required; 56*> = .FALSE.: only eigenvalues are required. 57*> \endverbatim 58*> 59*> \param[in] WANTZ 60*> \verbatim 61*> WANTZ is LOGICAL 62*> = .TRUE. : the matrix of Schur vectors Z is required; 63*> = .FALSE.: Schur vectors are not required. 64*> \endverbatim 65*> 66*> \param[in] N 67*> \verbatim 68*> N is INTEGER 69*> The order of the matrix H. N >= 0. 70*> \endverbatim 71*> 72*> \param[in] ILO 73*> \verbatim 74*> ILO is INTEGER 75*> \endverbatim 76*> 77*> \param[in] IHI 78*> \verbatim 79*> IHI is INTEGER 80*> 81*> It is assumed that H is already upper triangular in rows 82*> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, 83*> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a 84*> previous call to ZGEBAL, and then passed to ZGEHRD when the 85*> matrix output by ZGEBAL is reduced to Hessenberg form. 86*> Otherwise, ILO and IHI should be set to 1 and N, 87*> respectively. If N > 0, then 1 <= ILO <= IHI <= N. 88*> If N = 0, then ILO = 1 and IHI = 0. 89*> \endverbatim 90*> 91*> \param[in,out] H 92*> \verbatim 93*> H is COMPLEX*16 array, dimension (LDH,N) 94*> On entry, the upper Hessenberg matrix H. 95*> On exit, if INFO = 0 and WANTT is .TRUE., then H 96*> contains the upper triangular matrix T from the Schur 97*> decomposition (the Schur form). If INFO = 0 and WANT is 98*> .FALSE., then the contents of H are unspecified on exit. 99*> (The output value of H when INFO > 0 is given under the 100*> description of INFO below.) 101*> 102*> This subroutine may explicitly set H(i,j) = 0 for i > j and 103*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. 104*> \endverbatim 105*> 106*> \param[in] LDH 107*> \verbatim 108*> LDH is INTEGER 109*> The leading dimension of the array H. LDH >= max(1,N). 110*> \endverbatim 111*> 112*> \param[out] W 113*> \verbatim 114*> W is COMPLEX*16 array, dimension (N) 115*> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored 116*> in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are 117*> stored in the same order as on the diagonal of the Schur 118*> form returned in H, with W(i) = H(i,i). 119*> \endverbatim 120*> 121*> \param[in] ILOZ 122*> \verbatim 123*> ILOZ is INTEGER 124*> \endverbatim 125*> 126*> \param[in] IHIZ 127*> \verbatim 128*> IHIZ is INTEGER 129*> Specify the rows of Z to which transformations must be 130*> applied if WANTZ is .TRUE.. 131*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. 132*> \endverbatim 133*> 134*> \param[in,out] Z 135*> \verbatim 136*> Z is COMPLEX*16 array, dimension (LDZ,IHI) 137*> If WANTZ is .FALSE., then Z is not referenced. 138*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is 139*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the 140*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). 141*> (The output value of Z when INFO > 0 is given under 142*> the description of INFO below.) 143*> \endverbatim 144*> 145*> \param[in] LDZ 146*> \verbatim 147*> LDZ is INTEGER 148*> The leading dimension of the array Z. if WANTZ is .TRUE. 149*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. 150*> \endverbatim 151*> 152*> \param[out] WORK 153*> \verbatim 154*> WORK is COMPLEX*16 array, dimension LWORK 155*> On exit, if LWORK = -1, WORK(1) returns an estimate of 156*> the optimal value for LWORK. 157*> \endverbatim 158*> 159*> \param[in] LWORK 160*> \verbatim 161*> LWORK is INTEGER 162*> The dimension of the array WORK. LWORK >= max(1,N) 163*> is sufficient, but LWORK typically as large as 6*N may 164*> be required for optimal performance. A workspace query 165*> to determine the optimal workspace size is recommended. 166*> 167*> If LWORK = -1, then ZLAQR0 does a workspace query. 168*> In this case, ZLAQR0 checks the input parameters and 169*> estimates the optimal workspace size for the given 170*> values of N, ILO and IHI. The estimate is returned 171*> in WORK(1). No error message related to LWORK is 172*> issued by XERBLA. Neither H nor Z are accessed. 173*> \endverbatim 174*> 175*> \param[out] INFO 176*> \verbatim 177*> INFO is INTEGER 178*> = 0: successful exit 179*> > 0: if INFO = i, ZLAQR0 failed to compute all of 180*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR 181*> and WI contain those eigenvalues which have been 182*> successfully computed. (Failures are rare.) 183*> 184*> If INFO > 0 and WANT is .FALSE., then on exit, 185*> the remaining unconverged eigenvalues are the eigen- 186*> values of the upper Hessenberg matrix rows and 187*> columns ILO through INFO of the final, output 188*> value of H. 189*> 190*> If INFO > 0 and WANTT is .TRUE., then on exit 191*> 192*> (*) (initial value of H)*U = U*(final value of H) 193*> 194*> where U is a unitary matrix. The final 195*> value of H is upper Hessenberg and triangular in 196*> rows and columns INFO+1 through IHI. 197*> 198*> If INFO > 0 and WANTZ is .TRUE., then on exit 199*> 200*> (final value of Z(ILO:IHI,ILOZ:IHIZ) 201*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U 202*> 203*> where U is the unitary matrix in (*) (regard- 204*> less of the value of WANTT.) 205*> 206*> If INFO > 0 and WANTZ is .FALSE., then Z is not 207*> accessed. 208*> \endverbatim 209* 210* Authors: 211* ======== 212* 213*> \author Univ. of Tennessee 214*> \author Univ. of California Berkeley 215*> \author Univ. of Colorado Denver 216*> \author NAG Ltd. 217* 218*> \date December 2016 219* 220*> \ingroup complex16OTHERauxiliary 221* 222*> \par Contributors: 223* ================== 224*> 225*> Karen Braman and Ralph Byers, Department of Mathematics, 226*> University of Kansas, USA 227* 228*> \par References: 229* ================ 230*> 231*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 232*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 233*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages 234*> 929--947, 2002. 235*> \n 236*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 237*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal 238*> of Matrix Analysis, volume 23, pages 948--973, 2002. 239*> 240* ===================================================================== 241 SUBROUTINE ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 242 $ IHIZ, Z, LDZ, WORK, LWORK, INFO ) 243* 244* -- LAPACK auxiliary routine (version 3.7.0) -- 245* -- LAPACK is a software package provided by Univ. of Tennessee, -- 246* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 247* December 2016 248* 249* .. Scalar Arguments .. 250 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N 251 LOGICAL WANTT, WANTZ 252* .. 253* .. Array Arguments .. 254 COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) 255* .. 256* 257* ================================================================ 258* 259* .. Parameters .. 260* 261* ==== Matrices of order NTINY or smaller must be processed by 262* . ZLAHQR because of insufficient subdiagonal scratch space. 263* . (This is a hard limit.) ==== 264 INTEGER NTINY 265 PARAMETER ( NTINY = 11 ) 266* 267* ==== Exceptional deflation windows: try to cure rare 268* . slow convergence by varying the size of the 269* . deflation window after KEXNW iterations. ==== 270 INTEGER KEXNW 271 PARAMETER ( KEXNW = 5 ) 272* 273* ==== Exceptional shifts: try to cure rare slow convergence 274* . with ad-hoc exceptional shifts every KEXSH iterations. 275* . ==== 276 INTEGER KEXSH 277 PARAMETER ( KEXSH = 6 ) 278* 279* ==== The constant WILK1 is used to form the exceptional 280* . shifts. ==== 281 DOUBLE PRECISION WILK1 282 PARAMETER ( WILK1 = 0.75d0 ) 283 COMPLEX*16 ZERO, ONE 284 PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), 285 $ ONE = ( 1.0d0, 0.0d0 ) ) 286 DOUBLE PRECISION TWO 287 PARAMETER ( TWO = 2.0d0 ) 288* .. 289* .. Local Scalars .. 290 COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2 291 DOUBLE PRECISION S 292 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, 293 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, 294 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS, 295 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD 296 LOGICAL SORTED 297 CHARACTER JBCMPZ*2 298* .. 299* .. External Functions .. 300 INTEGER ILAENV 301 EXTERNAL ILAENV 302* .. 303* .. Local Arrays .. 304 COMPLEX*16 ZDUM( 1, 1 ) 305* .. 306* .. External Subroutines .. 307 EXTERNAL ZLACPY, ZLAHQR, ZLAQR3, ZLAQR4, ZLAQR5 308* .. 309* .. Intrinsic Functions .. 310 INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX, MIN, MOD, 311 $ SQRT 312* .. 313* .. Statement Functions .. 314 DOUBLE PRECISION CABS1 315* .. 316* .. Statement Function definitions .. 317 CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) ) 318* .. 319* .. Executable Statements .. 320 INFO = 0 321* 322* ==== Quick return for N = 0: nothing to do. ==== 323* 324 IF( N.EQ.0 ) THEN 325 WORK( 1 ) = ONE 326 RETURN 327 END IF 328* 329 IF( N.LE.NTINY ) THEN 330* 331* ==== Tiny matrices must use ZLAHQR. ==== 332* 333 LWKOPT = 1 334 IF( LWORK.NE.-1 ) 335 $ CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, 336 $ IHIZ, Z, LDZ, INFO ) 337 ELSE 338* 339* ==== Use small bulge multi-shift QR with aggressive early 340* . deflation on larger-than-tiny matrices. ==== 341* 342* ==== Hope for the best. ==== 343* 344 INFO = 0 345* 346* ==== Set up job flags for ILAENV. ==== 347* 348 IF( WANTT ) THEN 349 JBCMPZ( 1: 1 ) = 'S' 350 ELSE 351 JBCMPZ( 1: 1 ) = 'E' 352 END IF 353 IF( WANTZ ) THEN 354 JBCMPZ( 2: 2 ) = 'V' 355 ELSE 356 JBCMPZ( 2: 2 ) = 'N' 357 END IF 358* 359* ==== NWR = recommended deflation window size. At this 360* . point, N .GT. NTINY = 11, so there is enough 361* . subdiagonal workspace for NWR.GE.2 as required. 362* . (In fact, there is enough subdiagonal space for 363* . NWR.GE.3.) ==== 364* 365 NWR = ILAENV( 13, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 366 NWR = MAX( 2, NWR ) 367 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) 368* 369* ==== NSR = recommended number of simultaneous shifts. 370* . At this point N .GT. NTINY = 11, so there is at 371* . enough subdiagonal workspace for NSR to be even 372* . and greater than or equal to two as required. ==== 373* 374 NSR = ILAENV( 15, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 375 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO ) 376 NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) 377* 378* ==== Estimate optimal workspace ==== 379* 380* ==== Workspace query call to ZLAQR3 ==== 381* 382 CALL ZLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, 383 $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H, 384 $ LDH, WORK, -1 ) 385* 386* ==== Optimal workspace = MAX(ZLAQR5, ZLAQR3) ==== 387* 388 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) 389* 390* ==== Quick return in case of workspace query. ==== 391* 392 IF( LWORK.EQ.-1 ) THEN 393 WORK( 1 ) = DCMPLX( LWKOPT, 0 ) 394 RETURN 395 END IF 396* 397* ==== ZLAHQR/ZLAQR0 crossover point ==== 398* 399 NMIN = ILAENV( 12, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 400 NMIN = MAX( NTINY, NMIN ) 401* 402* ==== Nibble crossover point ==== 403* 404 NIBBLE = ILAENV( 14, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 405 NIBBLE = MAX( 0, NIBBLE ) 406* 407* ==== Accumulate reflections during ttswp? Use block 408* . 2-by-2 structure during matrix-matrix multiply? ==== 409* 410 KACC22 = ILAENV( 16, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 411 KACC22 = MAX( 0, KACC22 ) 412 KACC22 = MIN( 2, KACC22 ) 413* 414* ==== NWMAX = the largest possible deflation window for 415* . which there is sufficient workspace. ==== 416* 417 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) 418 NW = NWMAX 419* 420* ==== NSMAX = the Largest number of simultaneous shifts 421* . for which there is sufficient workspace. ==== 422* 423 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 ) 424 NSMAX = NSMAX - MOD( NSMAX, 2 ) 425* 426* ==== NDFL: an iteration count restarted at deflation. ==== 427* 428 NDFL = 1 429* 430* ==== ITMAX = iteration limit ==== 431* 432 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) 433* 434* ==== Last row and column in the active block ==== 435* 436 KBOT = IHI 437* 438* ==== Main Loop ==== 439* 440 DO 70 IT = 1, ITMAX 441* 442* ==== Done when KBOT falls below ILO ==== 443* 444 IF( KBOT.LT.ILO ) 445 $ GO TO 80 446* 447* ==== Locate active block ==== 448* 449 DO 10 K = KBOT, ILO + 1, -1 450 IF( H( K, K-1 ).EQ.ZERO ) 451 $ GO TO 20 452 10 CONTINUE 453 K = ILO 454 20 CONTINUE 455 KTOP = K 456* 457* ==== Select deflation window size: 458* . Typical Case: 459* . If possible and advisable, nibble the entire 460* . active block. If not, use size MIN(NWR,NWMAX) 461* . or MIN(NWR+1,NWMAX) depending upon which has 462* . the smaller corresponding subdiagonal entry 463* . (a heuristic). 464* . 465* . Exceptional Case: 466* . If there have been no deflations in KEXNW or 467* . more iterations, then vary the deflation window 468* . size. At first, because, larger windows are, 469* . in general, more powerful than smaller ones, 470* . rapidly increase the window to the maximum possible. 471* . Then, gradually reduce the window size. ==== 472* 473 NH = KBOT - KTOP + 1 474 NWUPBD = MIN( NH, NWMAX ) 475 IF( NDFL.LT.KEXNW ) THEN 476 NW = MIN( NWUPBD, NWR ) 477 ELSE 478 NW = MIN( NWUPBD, 2*NW ) 479 END IF 480 IF( NW.LT.NWMAX ) THEN 481 IF( NW.GE.NH-1 ) THEN 482 NW = NH 483 ELSE 484 KWTOP = KBOT - NW + 1 485 IF( CABS1( H( KWTOP, KWTOP-1 ) ).GT. 486 $ CABS1( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 487 END IF 488 END IF 489 IF( NDFL.LT.KEXNW ) THEN 490 NDEC = -1 491 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN 492 NDEC = NDEC + 1 493 IF( NW-NDEC.LT.2 ) 494 $ NDEC = 0 495 NW = NW - NDEC 496 END IF 497* 498* ==== Aggressive early deflation: 499* . split workspace under the subdiagonal into 500* . - an nw-by-nw work array V in the lower 501* . left-hand-corner, 502* . - an NW-by-at-least-NW-but-more-is-better 503* . (NW-by-NHO) horizontal work array along 504* . the bottom edge, 505* . - an at-least-NW-but-more-is-better (NHV-by-NW) 506* . vertical work array along the left-hand-edge. 507* . ==== 508* 509 KV = N - NW + 1 510 KT = NW + 1 511 NHO = ( N-NW-1 ) - KT + 1 512 KWV = NW + 2 513 NVE = ( N-NW ) - KWV + 1 514* 515* ==== Aggressive early deflation ==== 516* 517 CALL ZLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, 518 $ IHIZ, Z, LDZ, LS, LD, W, H( KV, 1 ), LDH, NHO, 519 $ H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, WORK, 520 $ LWORK ) 521* 522* ==== Adjust KBOT accounting for new deflations. ==== 523* 524 KBOT = KBOT - LD 525* 526* ==== KS points to the shifts. ==== 527* 528 KS = KBOT - LS + 1 529* 530* ==== Skip an expensive QR sweep if there is a (partly 531* . heuristic) reason to expect that many eigenvalues 532* . will deflate without it. Here, the QR sweep is 533* . skipped if many eigenvalues have just been deflated 534* . or if the remaining active block is small. 535* 536 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- 537 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN 538* 539* ==== NS = nominal number of simultaneous shifts. 540* . This may be lowered (slightly) if ZLAQR3 541* . did not provide that many shifts. ==== 542* 543 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) 544 NS = NS - MOD( NS, 2 ) 545* 546* ==== If there have been no deflations 547* . in a multiple of KEXSH iterations, 548* . then try exceptional shifts. 549* . Otherwise use shifts provided by 550* . ZLAQR3 above or from the eigenvalues 551* . of a trailing principal submatrix. ==== 552* 553 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN 554 KS = KBOT - NS + 1 555 DO 30 I = KBOT, KS + 1, -2 556 W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) ) 557 W( I-1 ) = W( I ) 558 30 CONTINUE 559 ELSE 560* 561* ==== Got NS/2 or fewer shifts? Use ZLAQR4 or 562* . ZLAHQR on a trailing principal submatrix to 563* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, 564* . there is enough space below the subdiagonal 565* . to fit an NS-by-NS scratch array.) ==== 566* 567 IF( KBOT-KS+1.LE.NS / 2 ) THEN 568 KS = KBOT - NS + 1 569 KT = N - NS + 1 570 CALL ZLACPY( 'A', NS, NS, H( KS, KS ), LDH, 571 $ H( KT, 1 ), LDH ) 572 IF( NS.GT.NMIN ) THEN 573 CALL ZLAQR4( .false., .false., NS, 1, NS, 574 $ H( KT, 1 ), LDH, W( KS ), 1, 1, 575 $ ZDUM, 1, WORK, LWORK, INF ) 576 ELSE 577 CALL ZLAHQR( .false., .false., NS, 1, NS, 578 $ H( KT, 1 ), LDH, W( KS ), 1, 1, 579 $ ZDUM, 1, INF ) 580 END IF 581 KS = KS + INF 582* 583* ==== In case of a rare QR failure use 584* . eigenvalues of the trailing 2-by-2 585* . principal submatrix. Scale to avoid 586* . overflows, underflows and subnormals. 587* . (The scale factor S can not be zero, 588* . because H(KBOT,KBOT-1) is nonzero.) ==== 589* 590 IF( KS.GE.KBOT ) THEN 591 S = CABS1( H( KBOT-1, KBOT-1 ) ) + 592 $ CABS1( H( KBOT, KBOT-1 ) ) + 593 $ CABS1( H( KBOT-1, KBOT ) ) + 594 $ CABS1( H( KBOT, KBOT ) ) 595 AA = H( KBOT-1, KBOT-1 ) / S 596 CC = H( KBOT, KBOT-1 ) / S 597 BB = H( KBOT-1, KBOT ) / S 598 DD = H( KBOT, KBOT ) / S 599 TR2 = ( AA+DD ) / TWO 600 DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC 601 RTDISC = SQRT( -DET ) 602 W( KBOT-1 ) = ( TR2+RTDISC )*S 603 W( KBOT ) = ( TR2-RTDISC )*S 604* 605 KS = KBOT - 1 606 END IF 607 END IF 608* 609 IF( KBOT-KS+1.GT.NS ) THEN 610* 611* ==== Sort the shifts (Helps a little) ==== 612* 613 SORTED = .false. 614 DO 50 K = KBOT, KS + 1, -1 615 IF( SORTED ) 616 $ GO TO 60 617 SORTED = .true. 618 DO 40 I = KS, K - 1 619 IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) ) 620 $ THEN 621 SORTED = .false. 622 SWAP = W( I ) 623 W( I ) = W( I+1 ) 624 W( I+1 ) = SWAP 625 END IF 626 40 CONTINUE 627 50 CONTINUE 628 60 CONTINUE 629 END IF 630 END IF 631* 632* ==== If there are only two shifts, then use 633* . only one. ==== 634* 635 IF( KBOT-KS+1.EQ.2 ) THEN 636 IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT. 637 $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN 638 W( KBOT-1 ) = W( KBOT ) 639 ELSE 640 W( KBOT ) = W( KBOT-1 ) 641 END IF 642 END IF 643* 644* ==== Use up to NS of the the smallest magnitude 645* . shifts. If there aren't NS shifts available, 646* . then use them all, possibly dropping one to 647* . make the number of shifts even. ==== 648* 649 NS = MIN( NS, KBOT-KS+1 ) 650 NS = NS - MOD( NS, 2 ) 651 KS = KBOT - NS + 1 652* 653* ==== Small-bulge multi-shift QR sweep: 654* . split workspace under the subdiagonal into 655* . - a KDU-by-KDU work array U in the lower 656* . left-hand-corner, 657* . - a KDU-by-at-least-KDU-but-more-is-better 658* . (KDU-by-NHo) horizontal work array WH along 659* . the bottom edge, 660* . - and an at-least-KDU-but-more-is-better-by-KDU 661* . (NVE-by-KDU) vertical work WV arrow along 662* . the left-hand-edge. ==== 663* 664 KDU = 3*NS - 3 665 KU = N - KDU + 1 666 KWH = KDU + 1 667 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 668 KWV = KDU + 4 669 NVE = N - KDU - KWV + 1 670* 671* ==== Small-bulge multi-shift QR sweep ==== 672* 673 CALL ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, 674 $ W( KS ), H, LDH, ILOZ, IHIZ, Z, LDZ, WORK, 675 $ 3, H( KU, 1 ), LDH, NVE, H( KWV, 1 ), LDH, 676 $ NHO, H( KU, KWH ), LDH ) 677 END IF 678* 679* ==== Note progress (or the lack of it). ==== 680* 681 IF( LD.GT.0 ) THEN 682 NDFL = 1 683 ELSE 684 NDFL = NDFL + 1 685 END IF 686* 687* ==== End of main loop ==== 688 70 CONTINUE 689* 690* ==== Iteration limit exceeded. Set INFO to show where 691* . the problem occurred and exit. ==== 692* 693 INFO = KBOT 694 80 CONTINUE 695 END IF 696* 697* ==== Return the optimal value of LWORK. ==== 698* 699 WORK( 1 ) = DCMPLX( LWKOPT, 0 ) 700* 701* ==== End of ZLAQR0 ==== 702* 703 END 704