1*> \brief \b DLAQR0 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 DLAQR0 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr0.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr0.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr0.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, 22* ILOZ, 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* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), 30* $ Z( LDZ, * ) 31* .. 32* 33* 34*> \par Purpose: 35* ============= 36*> 37*> \verbatim 38*> 39*> DLAQR0 computes the eigenvalues of a Hessenberg matrix H 40*> and, optionally, the matrices T and Z from the Schur decomposition 41*> H = Z T Z**T, where T is an upper quasi-triangular matrix (the 42*> Schur form), and Z is the orthogonal matrix of Schur vectors. 43*> 44*> Optionally Z may be postmultiplied into an input orthogonal 45*> matrix Q so that this routine can give the Schur factorization 46*> of a matrix A which has been reduced to the Hessenberg form H 47*> by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. 48*> \endverbatim 49* 50* Arguments: 51* ========== 52* 53*> \param[in] WANTT 54*> \verbatim 55*> WANTT is LOGICAL 56*> = .TRUE. : the full Schur form T is required; 57*> = .FALSE.: only eigenvalues are required. 58*> \endverbatim 59*> 60*> \param[in] WANTZ 61*> \verbatim 62*> WANTZ is LOGICAL 63*> = .TRUE. : the matrix of Schur vectors Z is required; 64*> = .FALSE.: Schur vectors are not required. 65*> \endverbatim 66*> 67*> \param[in] N 68*> \verbatim 69*> N is INTEGER 70*> The order of the matrix H. N >= 0. 71*> \endverbatim 72*> 73*> \param[in] ILO 74*> \verbatim 75*> ILO is INTEGER 76*> \endverbatim 77*> 78*> \param[in] IHI 79*> \verbatim 80*> IHI is INTEGER 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 DGEBAL, and then passed to DGEHRD when the 85*> matrix output by DGEBAL 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 DOUBLE PRECISION array, dimension (LDH,N) 94*> On entry, the upper Hessenberg matrix H. 95*> On exit, if INFO = 0 and WANTT is .TRUE., then H contains 96*> the upper quasi-triangular matrix T from the Schur 97*> decomposition (the Schur form); 2-by-2 diagonal blocks 98*> (corresponding to complex conjugate pairs of eigenvalues) 99*> are returned in standard form, with H(i,i) = H(i+1,i+1) 100*> and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is 101*> .FALSE., then the contents of H are unspecified on exit. 102*> (The output value of H when INFO > 0 is given under the 103*> description of INFO below.) 104*> 105*> This subroutine may explicitly set H(i,j) = 0 for i > j and 106*> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. 107*> \endverbatim 108*> 109*> \param[in] LDH 110*> \verbatim 111*> LDH is INTEGER 112*> The leading dimension of the array H. LDH >= max(1,N). 113*> \endverbatim 114*> 115*> \param[out] WR 116*> \verbatim 117*> WR is DOUBLE PRECISION array, dimension (IHI) 118*> \endverbatim 119*> 120*> \param[out] WI 121*> \verbatim 122*> WI is DOUBLE PRECISION array, dimension (IHI) 123*> The real and imaginary parts, respectively, of the computed 124*> eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) 125*> and WI(ILO:IHI). If two eigenvalues are computed as a 126*> complex conjugate pair, they are stored in consecutive 127*> elements of WR and WI, say the i-th and (i+1)th, with 128*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then 129*> the eigenvalues are stored in the same order as on the 130*> diagonal of the Schur form returned in H, with 131*> WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal 132*> block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and 133*> WI(i+1) = -WI(i). 134*> \endverbatim 135*> 136*> \param[in] ILOZ 137*> \verbatim 138*> ILOZ is INTEGER 139*> \endverbatim 140*> 141*> \param[in] IHIZ 142*> \verbatim 143*> IHIZ is INTEGER 144*> Specify the rows of Z to which transformations must be 145*> applied if WANTZ is .TRUE.. 146*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. 147*> \endverbatim 148*> 149*> \param[in,out] Z 150*> \verbatim 151*> Z is DOUBLE PRECISION array, dimension (LDZ,IHI) 152*> If WANTZ is .FALSE., then Z is not referenced. 153*> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is 154*> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the 155*> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). 156*> (The output value of Z when INFO > 0 is given under 157*> the description of INFO below.) 158*> \endverbatim 159*> 160*> \param[in] LDZ 161*> \verbatim 162*> LDZ is INTEGER 163*> The leading dimension of the array Z. if WANTZ is .TRUE. 164*> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. 165*> \endverbatim 166*> 167*> \param[out] WORK 168*> \verbatim 169*> WORK is DOUBLE PRECISION array, dimension LWORK 170*> On exit, if LWORK = -1, WORK(1) returns an estimate of 171*> the optimal value for LWORK. 172*> \endverbatim 173*> 174*> \param[in] LWORK 175*> \verbatim 176*> LWORK is INTEGER 177*> The dimension of the array WORK. LWORK >= max(1,N) 178*> is sufficient, but LWORK typically as large as 6*N may 179*> be required for optimal performance. A workspace query 180*> to determine the optimal workspace size is recommended. 181*> 182*> If LWORK = -1, then DLAQR0 does a workspace query. 183*> In this case, DLAQR0 checks the input parameters and 184*> estimates the optimal workspace size for the given 185*> values of N, ILO and IHI. The estimate is returned 186*> in WORK(1). No error message related to LWORK is 187*> issued by XERBLA. Neither H nor Z are accessed. 188*> \endverbatim 189*> 190*> \param[out] INFO 191*> \verbatim 192*> INFO is INTEGER 193*> = 0: successful exit 194*> > 0: if INFO = i, DLAQR0 failed to compute all of 195*> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR 196*> and WI contain those eigenvalues which have been 197*> successfully computed. (Failures are rare.) 198*> 199*> If INFO > 0 and WANT is .FALSE., then on exit, 200*> the remaining unconverged eigenvalues are the eigen- 201*> values of the upper Hessenberg matrix rows and 202*> columns ILO through INFO of the final, output 203*> value of H. 204*> 205*> If INFO > 0 and WANTT is .TRUE., then on exit 206*> 207*> (*) (initial value of H)*U = U*(final value of H) 208*> 209*> where U is an orthogonal matrix. The final 210*> value of H is upper Hessenberg and quasi-triangular 211*> in rows and columns INFO+1 through IHI. 212*> 213*> If INFO > 0 and WANTZ is .TRUE., then on exit 214*> 215*> (final value of Z(ILO:IHI,ILOZ:IHIZ) 216*> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U 217*> 218*> where U is the orthogonal matrix in (*) (regard- 219*> less of the value of WANTT.) 220*> 221*> If INFO > 0 and WANTZ is .FALSE., then Z is not 222*> accessed. 223*> \endverbatim 224* 225*> \par Contributors: 226* ================== 227*> 228*> Karen Braman and Ralph Byers, Department of Mathematics, 229*> University of Kansas, USA 230* 231*> \par References: 232* ================ 233*> 234*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 235*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 236*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages 237*> 929--947, 2002. 238*> \n 239*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR 240*> Algorithm Part II: Aggressive Early Deflation, SIAM Journal 241*> of Matrix Analysis, volume 23, pages 948--973, 2002. 242* 243* Authors: 244* ======== 245* 246*> \author Univ. of Tennessee 247*> \author Univ. of California Berkeley 248*> \author Univ. of Colorado Denver 249*> \author NAG Ltd. 250* 251*> \ingroup doubleOTHERauxiliary 252* 253* ===================================================================== 254 SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, 255 $ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) 256* 257* -- LAPACK auxiliary routine -- 258* -- LAPACK is a software package provided by Univ. of Tennessee, -- 259* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 260* 261* .. Scalar Arguments .. 262 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N 263 LOGICAL WANTT, WANTZ 264* .. 265* .. Array Arguments .. 266 DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), 267 $ Z( LDZ, * ) 268* .. 269* 270* ================================================================ 271* 272* .. Parameters .. 273* 274* ==== Matrices of order NTINY or smaller must be processed by 275* . DLAHQR because of insufficient subdiagonal scratch space. 276* . (This is a hard limit.) ==== 277 INTEGER NTINY 278 PARAMETER ( NTINY = 15 ) 279* 280* ==== Exceptional deflation windows: try to cure rare 281* . slow convergence by varying the size of the 282* . deflation window after KEXNW iterations. ==== 283 INTEGER KEXNW 284 PARAMETER ( KEXNW = 5 ) 285* 286* ==== Exceptional shifts: try to cure rare slow convergence 287* . with ad-hoc exceptional shifts every KEXSH iterations. 288* . ==== 289 INTEGER KEXSH 290 PARAMETER ( KEXSH = 6 ) 291* 292* ==== The constants WILK1 and WILK2 are used to form the 293* . exceptional shifts. ==== 294 DOUBLE PRECISION WILK1, WILK2 295 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 ) 296 DOUBLE PRECISION ZERO, ONE 297 PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 ) 298* .. 299* .. Local Scalars .. 300 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP 301 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, 302 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, 303 $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS, 304 $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD 305 LOGICAL SORTED 306 CHARACTER JBCMPZ*2 307* .. 308* .. External Functions .. 309 INTEGER ILAENV 310 EXTERNAL ILAENV 311* .. 312* .. Local Arrays .. 313 DOUBLE PRECISION ZDUM( 1, 1 ) 314* .. 315* .. External Subroutines .. 316 EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, DLAQR5 317* .. 318* .. Intrinsic Functions .. 319 INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD 320* .. 321* .. Executable Statements .. 322 INFO = 0 323* 324* ==== Quick return for N = 0: nothing to do. ==== 325* 326 IF( N.EQ.0 ) THEN 327 WORK( 1 ) = ONE 328 RETURN 329 END IF 330* 331 IF( N.LE.NTINY ) THEN 332* 333* ==== Tiny matrices must use DLAHQR. ==== 334* 335 LWKOPT = 1 336 IF( LWORK.NE.-1 ) 337 $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, 338 $ ILOZ, IHIZ, Z, LDZ, INFO ) 339 ELSE 340* 341* ==== Use small bulge multi-shift QR with aggressive early 342* . deflation on larger-than-tiny matrices. ==== 343* 344* ==== Hope for the best. ==== 345* 346 INFO = 0 347* 348* ==== Set up job flags for ILAENV. ==== 349* 350 IF( WANTT ) THEN 351 JBCMPZ( 1: 1 ) = 'S' 352 ELSE 353 JBCMPZ( 1: 1 ) = 'E' 354 END IF 355 IF( WANTZ ) THEN 356 JBCMPZ( 2: 2 ) = 'V' 357 ELSE 358 JBCMPZ( 2: 2 ) = 'N' 359 END IF 360* 361* ==== NWR = recommended deflation window size. At this 362* . point, N .GT. NTINY = 15, so there is enough 363* . subdiagonal workspace for NWR.GE.2 as required. 364* . (In fact, there is enough subdiagonal space for 365* . NWR.GE.4.) ==== 366* 367 NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 368 NWR = MAX( 2, NWR ) 369 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) 370* 371* ==== NSR = recommended number of simultaneous shifts. 372* . At this point N .GT. NTINY = 15, so there is at 373* . enough subdiagonal workspace for NSR to be even 374* . and greater than or equal to two as required. ==== 375* 376 NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 377 NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO ) 378 NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) 379* 380* ==== Estimate optimal workspace ==== 381* 382* ==== Workspace query call to DLAQR3 ==== 383* 384 CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, 385 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH, 386 $ N, H, LDH, WORK, -1 ) 387* 388* ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ==== 389* 390 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) 391* 392* ==== Quick return in case of workspace query. ==== 393* 394 IF( LWORK.EQ.-1 ) THEN 395 WORK( 1 ) = DBLE( LWKOPT ) 396 RETURN 397 END IF 398* 399* ==== DLAHQR/DLAQR0 crossover point ==== 400* 401 NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 402 NMIN = MAX( NTINY, NMIN ) 403* 404* ==== Nibble crossover point ==== 405* 406 NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 407 NIBBLE = MAX( 0, NIBBLE ) 408* 409* ==== Accumulate reflections during ttswp? Use block 410* . 2-by-2 structure during matrix-matrix multiply? ==== 411* 412 KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) 413 KACC22 = MAX( 0, KACC22 ) 414 KACC22 = MIN( 2, KACC22 ) 415* 416* ==== NWMAX = the largest possible deflation window for 417* . which there is sufficient workspace. ==== 418* 419 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) 420 NW = NWMAX 421* 422* ==== NSMAX = the Largest number of simultaneous shifts 423* . for which there is sufficient workspace. ==== 424* 425 NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 ) 426 NSMAX = NSMAX - MOD( NSMAX, 2 ) 427* 428* ==== NDFL: an iteration count restarted at deflation. ==== 429* 430 NDFL = 1 431* 432* ==== ITMAX = iteration limit ==== 433* 434 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) 435* 436* ==== Last row and column in the active block ==== 437* 438 KBOT = IHI 439* 440* ==== Main Loop ==== 441* 442 DO 80 IT = 1, ITMAX 443* 444* ==== Done when KBOT falls below ILO ==== 445* 446 IF( KBOT.LT.ILO ) 447 $ GO TO 90 448* 449* ==== Locate active block ==== 450* 451 DO 10 K = KBOT, ILO + 1, -1 452 IF( H( K, K-1 ).EQ.ZERO ) 453 $ GO TO 20 454 10 CONTINUE 455 K = ILO 456 20 CONTINUE 457 KTOP = K 458* 459* ==== Select deflation window size: 460* . Typical Case: 461* . If possible and advisable, nibble the entire 462* . active block. If not, use size MIN(NWR,NWMAX) 463* . or MIN(NWR+1,NWMAX) depending upon which has 464* . the smaller corresponding subdiagonal entry 465* . (a heuristic). 466* . 467* . Exceptional Case: 468* . If there have been no deflations in KEXNW or 469* . more iterations, then vary the deflation window 470* . size. At first, because, larger windows are, 471* . in general, more powerful than smaller ones, 472* . rapidly increase the window to the maximum possible. 473* . Then, gradually reduce the window size. ==== 474* 475 NH = KBOT - KTOP + 1 476 NWUPBD = MIN( NH, NWMAX ) 477 IF( NDFL.LT.KEXNW ) THEN 478 NW = MIN( NWUPBD, NWR ) 479 ELSE 480 NW = MIN( NWUPBD, 2*NW ) 481 END IF 482 IF( NW.LT.NWMAX ) THEN 483 IF( NW.GE.NH-1 ) THEN 484 NW = NH 485 ELSE 486 KWTOP = KBOT - NW + 1 487 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT. 488 $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 489 END IF 490 END IF 491 IF( NDFL.LT.KEXNW ) THEN 492 NDEC = -1 493 ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN 494 NDEC = NDEC + 1 495 IF( NW-NDEC.LT.2 ) 496 $ NDEC = 0 497 NW = NW - NDEC 498 END IF 499* 500* ==== Aggressive early deflation: 501* . split workspace under the subdiagonal into 502* . - an nw-by-nw work array V in the lower 503* . left-hand-corner, 504* . - an NW-by-at-least-NW-but-more-is-better 505* . (NW-by-NHO) horizontal work array along 506* . the bottom edge, 507* . - an at-least-NW-but-more-is-better (NHV-by-NW) 508* . vertical work array along the left-hand-edge. 509* . ==== 510* 511 KV = N - NW + 1 512 KT = NW + 1 513 NHO = ( N-NW-1 ) - KT + 1 514 KWV = NW + 2 515 NVE = ( N-NW ) - KWV + 1 516* 517* ==== Aggressive early deflation ==== 518* 519 CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, 520 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH, 521 $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, 522 $ WORK, LWORK ) 523* 524* ==== Adjust KBOT accounting for new deflations. ==== 525* 526 KBOT = KBOT - LD 527* 528* ==== KS points to the shifts. ==== 529* 530 KS = KBOT - LS + 1 531* 532* ==== Skip an expensive QR sweep if there is a (partly 533* . heuristic) reason to expect that many eigenvalues 534* . will deflate without it. Here, the QR sweep is 535* . skipped if many eigenvalues have just been deflated 536* . or if the remaining active block is small. 537* 538 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- 539 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN 540* 541* ==== NS = nominal number of simultaneous shifts. 542* . This may be lowered (slightly) if DLAQR3 543* . did not provide that many shifts. ==== 544* 545 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) 546 NS = NS - MOD( NS, 2 ) 547* 548* ==== If there have been no deflations 549* . in a multiple of KEXSH iterations, 550* . then try exceptional shifts. 551* . Otherwise use shifts provided by 552* . DLAQR3 above or from the eigenvalues 553* . of a trailing principal submatrix. ==== 554* 555 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN 556 KS = KBOT - NS + 1 557 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2 558 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) 559 AA = WILK1*SS + H( I, I ) 560 BB = SS 561 CC = WILK2*SS 562 DD = AA 563 CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ), 564 $ WR( I ), WI( I ), CS, SN ) 565 30 CONTINUE 566 IF( KS.EQ.KTOP ) THEN 567 WR( KS+1 ) = H( KS+1, KS+1 ) 568 WI( KS+1 ) = ZERO 569 WR( KS ) = WR( KS+1 ) 570 WI( KS ) = WI( KS+1 ) 571 END IF 572 ELSE 573* 574* ==== Got NS/2 or fewer shifts? Use DLAQR4 or 575* . DLAHQR on a trailing principal submatrix to 576* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6, 577* . there is enough space below the subdiagonal 578* . to fit an NS-by-NS scratch array.) ==== 579* 580 IF( KBOT-KS+1.LE.NS / 2 ) THEN 581 KS = KBOT - NS + 1 582 KT = N - NS + 1 583 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH, 584 $ H( KT, 1 ), LDH ) 585 IF( NS.GT.NMIN ) THEN 586 CALL DLAQR4( .false., .false., NS, 1, NS, 587 $ H( KT, 1 ), LDH, WR( KS ), 588 $ WI( KS ), 1, 1, ZDUM, 1, WORK, 589 $ LWORK, INF ) 590 ELSE 591 CALL DLAHQR( .false., .false., NS, 1, NS, 592 $ H( KT, 1 ), LDH, WR( KS ), 593 $ WI( KS ), 1, 1, ZDUM, 1, INF ) 594 END IF 595 KS = KS + INF 596* 597* ==== In case of a rare QR failure use 598* . eigenvalues of the trailing 2-by-2 599* . principal submatrix. ==== 600* 601 IF( KS.GE.KBOT ) THEN 602 AA = H( KBOT-1, KBOT-1 ) 603 CC = H( KBOT, KBOT-1 ) 604 BB = H( KBOT-1, KBOT ) 605 DD = H( KBOT, KBOT ) 606 CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ), 607 $ WI( KBOT-1 ), WR( KBOT ), 608 $ WI( KBOT ), CS, SN ) 609 KS = KBOT - 1 610 END IF 611 END IF 612* 613 IF( KBOT-KS+1.GT.NS ) THEN 614* 615* ==== Sort the shifts (Helps a little) 616* . Bubble sort keeps complex conjugate 617* . pairs together. ==== 618* 619 SORTED = .false. 620 DO 50 K = KBOT, KS + 1, -1 621 IF( SORTED ) 622 $ GO TO 60 623 SORTED = .true. 624 DO 40 I = KS, K - 1 625 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT. 626 $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN 627 SORTED = .false. 628* 629 SWAP = WR( I ) 630 WR( I ) = WR( I+1 ) 631 WR( I+1 ) = SWAP 632* 633 SWAP = WI( I ) 634 WI( I ) = WI( I+1 ) 635 WI( I+1 ) = SWAP 636 END IF 637 40 CONTINUE 638 50 CONTINUE 639 60 CONTINUE 640 END IF 641* 642* ==== Shuffle shifts into pairs of real shifts 643* . and pairs of complex conjugate shifts 644* . assuming complex conjugate shifts are 645* . already adjacent to one another. (Yes, 646* . they are.) ==== 647* 648 DO 70 I = KBOT, KS + 2, -2 649 IF( WI( I ).NE.-WI( I-1 ) ) THEN 650* 651 SWAP = WR( I ) 652 WR( I ) = WR( I-1 ) 653 WR( I-1 ) = WR( I-2 ) 654 WR( I-2 ) = SWAP 655* 656 SWAP = WI( I ) 657 WI( I ) = WI( I-1 ) 658 WI( I-1 ) = WI( I-2 ) 659 WI( I-2 ) = SWAP 660 END IF 661 70 CONTINUE 662 END IF 663* 664* ==== If there are only two shifts and both are 665* . real, then use only one. ==== 666* 667 IF( KBOT-KS+1.EQ.2 ) THEN 668 IF( WI( KBOT ).EQ.ZERO ) THEN 669 IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT. 670 $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN 671 WR( KBOT-1 ) = WR( KBOT ) 672 ELSE 673 WR( KBOT ) = WR( KBOT-1 ) 674 END IF 675 END IF 676 END IF 677* 678* ==== Use up to NS of the the smallest magnitude 679* . shifts. If there aren't NS shifts available, 680* . then use them all, possibly dropping one to 681* . make the number of shifts even. ==== 682* 683 NS = MIN( NS, KBOT-KS+1 ) 684 NS = NS - MOD( NS, 2 ) 685 KS = KBOT - NS + 1 686* 687* ==== Small-bulge multi-shift QR sweep: 688* . split workspace under the subdiagonal into 689* . - a KDU-by-KDU work array U in the lower 690* . left-hand-corner, 691* . - a KDU-by-at-least-KDU-but-more-is-better 692* . (KDU-by-NHo) horizontal work array WH along 693* . the bottom edge, 694* . - and an at-least-KDU-but-more-is-better-by-KDU 695* . (NVE-by-KDU) vertical work WV arrow along 696* . the left-hand-edge. ==== 697* 698 KDU = 2*NS 699 KU = N - KDU + 1 700 KWH = KDU + 1 701 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 702 KWV = KDU + 4 703 NVE = N - KDU - KWV + 1 704* 705* ==== Small-bulge multi-shift QR sweep ==== 706* 707 CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, 708 $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z, 709 $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE, 710 $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH ) 711 END IF 712* 713* ==== Note progress (or the lack of it). ==== 714* 715 IF( LD.GT.0 ) THEN 716 NDFL = 1 717 ELSE 718 NDFL = NDFL + 1 719 END IF 720* 721* ==== End of main loop ==== 722 80 CONTINUE 723* 724* ==== Iteration limit exceeded. Set INFO to show where 725* . the problem occurred and exit. ==== 726* 727 INFO = KBOT 728 90 CONTINUE 729 END IF 730* 731* ==== Return the optimal value of LWORK. ==== 732* 733 WORK( 1 ) = DBLE( LWKOPT ) 734* 735* ==== End of DLAQR0 ==== 736* 737 END 738