Annotation of rpl/lapack/lapack/dlaqr3.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
! 2: $ IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
! 3: $ LDT, NV, WV, LDWV, WORK, LWORK )
! 4: *
! 5: * -- LAPACK auxiliary routine (version 3.2.1) --
! 6: * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
! 7: * -- April 2009 --
! 8: *
! 9: * .. Scalar Arguments ..
! 10: INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
! 11: $ LDZ, LWORK, N, ND, NH, NS, NV, NW
! 12: LOGICAL WANTT, WANTZ
! 13: * ..
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
! 16: $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
! 17: $ Z( LDZ, * )
! 18: * ..
! 19: *
! 20: * ******************************************************************
! 21: * Aggressive early deflation:
! 22: *
! 23: * This subroutine accepts as input an upper Hessenberg matrix
! 24: * H and performs an orthogonal similarity transformation
! 25: * designed to detect and deflate fully converged eigenvalues from
! 26: * a trailing principal submatrix. On output H has been over-
! 27: * written by a new Hessenberg matrix that is a perturbation of
! 28: * an orthogonal similarity transformation of H. It is to be
! 29: * hoped that the final version of H has many zero subdiagonal
! 30: * entries.
! 31: *
! 32: * ******************************************************************
! 33: * WANTT (input) LOGICAL
! 34: * If .TRUE., then the Hessenberg matrix H is fully updated
! 35: * so that the quasi-triangular Schur factor may be
! 36: * computed (in cooperation with the calling subroutine).
! 37: * If .FALSE., then only enough of H is updated to preserve
! 38: * the eigenvalues.
! 39: *
! 40: * WANTZ (input) LOGICAL
! 41: * If .TRUE., then the orthogonal matrix Z is updated so
! 42: * so that the orthogonal Schur factor may be computed
! 43: * (in cooperation with the calling subroutine).
! 44: * If .FALSE., then Z is not referenced.
! 45: *
! 46: * N (input) INTEGER
! 47: * The order of the matrix H and (if WANTZ is .TRUE.) the
! 48: * order of the orthogonal matrix Z.
! 49: *
! 50: * KTOP (input) INTEGER
! 51: * It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
! 52: * KBOT and KTOP together determine an isolated block
! 53: * along the diagonal of the Hessenberg matrix.
! 54: *
! 55: * KBOT (input) INTEGER
! 56: * It is assumed without a check that either
! 57: * KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
! 58: * determine an isolated block along the diagonal of the
! 59: * Hessenberg matrix.
! 60: *
! 61: * NW (input) INTEGER
! 62: * Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
! 63: *
! 64: * H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
! 65: * On input the initial N-by-N section of H stores the
! 66: * Hessenberg matrix undergoing aggressive early deflation.
! 67: * On output H has been transformed by an orthogonal
! 68: * similarity transformation, perturbed, and the returned
! 69: * to Hessenberg form that (it is to be hoped) has some
! 70: * zero subdiagonal entries.
! 71: *
! 72: * LDH (input) integer
! 73: * Leading dimension of H just as declared in the calling
! 74: * subroutine. N .LE. LDH
! 75: *
! 76: * ILOZ (input) INTEGER
! 77: * IHIZ (input) INTEGER
! 78: * Specify the rows of Z to which transformations must be
! 79: * applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
! 80: *
! 81: * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
! 82: * IF WANTZ is .TRUE., then on output, the orthogonal
! 83: * similarity transformation mentioned above has been
! 84: * accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
! 85: * If WANTZ is .FALSE., then Z is unreferenced.
! 86: *
! 87: * LDZ (input) integer
! 88: * The leading dimension of Z just as declared in the
! 89: * calling subroutine. 1 .LE. LDZ.
! 90: *
! 91: * NS (output) integer
! 92: * The number of unconverged (ie approximate) eigenvalues
! 93: * returned in SR and SI that may be used as shifts by the
! 94: * calling subroutine.
! 95: *
! 96: * ND (output) integer
! 97: * The number of converged eigenvalues uncovered by this
! 98: * subroutine.
! 99: *
! 100: * SR (output) DOUBLE PRECISION array, dimension KBOT
! 101: * SI (output) DOUBLE PRECISION array, dimension KBOT
! 102: * On output, the real and imaginary parts of approximate
! 103: * eigenvalues that may be used for shifts are stored in
! 104: * SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
! 105: * SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
! 106: * The real and imaginary parts of converged eigenvalues
! 107: * are stored in SR(KBOT-ND+1) through SR(KBOT) and
! 108: * SI(KBOT-ND+1) through SI(KBOT), respectively.
! 109: *
! 110: * V (workspace) DOUBLE PRECISION array, dimension (LDV,NW)
! 111: * An NW-by-NW work array.
! 112: *
! 113: * LDV (input) integer scalar
! 114: * The leading dimension of V just as declared in the
! 115: * calling subroutine. NW .LE. LDV
! 116: *
! 117: * NH (input) integer scalar
! 118: * The number of columns of T. NH.GE.NW.
! 119: *
! 120: * T (workspace) DOUBLE PRECISION array, dimension (LDT,NW)
! 121: *
! 122: * LDT (input) integer
! 123: * The leading dimension of T just as declared in the
! 124: * calling subroutine. NW .LE. LDT
! 125: *
! 126: * NV (input) integer
! 127: * The number of rows of work array WV available for
! 128: * workspace. NV.GE.NW.
! 129: *
! 130: * WV (workspace) DOUBLE PRECISION array, dimension (LDWV,NW)
! 131: *
! 132: * LDWV (input) integer
! 133: * The leading dimension of W just as declared in the
! 134: * calling subroutine. NW .LE. LDV
! 135: *
! 136: * WORK (workspace) DOUBLE PRECISION array, dimension LWORK.
! 137: * On exit, WORK(1) is set to an estimate of the optimal value
! 138: * of LWORK for the given values of N, NW, KTOP and KBOT.
! 139: *
! 140: * LWORK (input) integer
! 141: * The dimension of the work array WORK. LWORK = 2*NW
! 142: * suffices, but greater efficiency may result from larger
! 143: * values of LWORK.
! 144: *
! 145: * If LWORK = -1, then a workspace query is assumed; DLAQR3
! 146: * only estimates the optimal workspace size for the given
! 147: * values of N, NW, KTOP and KBOT. The estimate is returned
! 148: * in WORK(1). No error message related to LWORK is issued
! 149: * by XERBLA. Neither H nor Z are accessed.
! 150: *
! 151: * ================================================================
! 152: * Based on contributions by
! 153: * Karen Braman and Ralph Byers, Department of Mathematics,
! 154: * University of Kansas, USA
! 155: *
! 156: * ================================================================
! 157: * .. Parameters ..
! 158: DOUBLE PRECISION ZERO, ONE
! 159: PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
! 160: * ..
! 161: * .. Local Scalars ..
! 162: DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
! 163: $ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
! 164: INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
! 165: $ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
! 166: $ LWKOPT, NMIN
! 167: LOGICAL BULGE, SORTED
! 168: * ..
! 169: * .. External Functions ..
! 170: DOUBLE PRECISION DLAMCH
! 171: INTEGER ILAENV
! 172: EXTERNAL DLAMCH, ILAENV
! 173: * ..
! 174: * .. External Subroutines ..
! 175: EXTERNAL DCOPY, DGEHRD, DGEMM, DLABAD, DLACPY, DLAHQR,
! 176: $ DLANV2, DLAQR4, DLARF, DLARFG, DLASET, DORMHR,
! 177: $ DTREXC
! 178: * ..
! 179: * .. Intrinsic Functions ..
! 180: INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
! 181: * ..
! 182: * .. Executable Statements ..
! 183: *
! 184: * ==== Estimate optimal workspace. ====
! 185: *
! 186: JW = MIN( NW, KBOT-KTOP+1 )
! 187: IF( JW.LE.2 ) THEN
! 188: LWKOPT = 1
! 189: ELSE
! 190: *
! 191: * ==== Workspace query call to DGEHRD ====
! 192: *
! 193: CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
! 194: LWK1 = INT( WORK( 1 ) )
! 195: *
! 196: * ==== Workspace query call to DORMHR ====
! 197: *
! 198: CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
! 199: $ WORK, -1, INFO )
! 200: LWK2 = INT( WORK( 1 ) )
! 201: *
! 202: * ==== Workspace query call to DLAQR4 ====
! 203: *
! 204: CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR, SI, 1, JW,
! 205: $ V, LDV, WORK, -1, INFQR )
! 206: LWK3 = INT( WORK( 1 ) )
! 207: *
! 208: * ==== Optimal workspace ====
! 209: *
! 210: LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
! 211: END IF
! 212: *
! 213: * ==== Quick return in case of workspace query. ====
! 214: *
! 215: IF( LWORK.EQ.-1 ) THEN
! 216: WORK( 1 ) = DBLE( LWKOPT )
! 217: RETURN
! 218: END IF
! 219: *
! 220: * ==== Nothing to do ...
! 221: * ... for an empty active block ... ====
! 222: NS = 0
! 223: ND = 0
! 224: WORK( 1 ) = ONE
! 225: IF( KTOP.GT.KBOT )
! 226: $ RETURN
! 227: * ... nor for an empty deflation window. ====
! 228: IF( NW.LT.1 )
! 229: $ RETURN
! 230: *
! 231: * ==== Machine constants ====
! 232: *
! 233: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
! 234: SAFMAX = ONE / SAFMIN
! 235: CALL DLABAD( SAFMIN, SAFMAX )
! 236: ULP = DLAMCH( 'PRECISION' )
! 237: SMLNUM = SAFMIN*( DBLE( N ) / ULP )
! 238: *
! 239: * ==== Setup deflation window ====
! 240: *
! 241: JW = MIN( NW, KBOT-KTOP+1 )
! 242: KWTOP = KBOT - JW + 1
! 243: IF( KWTOP.EQ.KTOP ) THEN
! 244: S = ZERO
! 245: ELSE
! 246: S = H( KWTOP, KWTOP-1 )
! 247: END IF
! 248: *
! 249: IF( KBOT.EQ.KWTOP ) THEN
! 250: *
! 251: * ==== 1-by-1 deflation window: not much to do ====
! 252: *
! 253: SR( KWTOP ) = H( KWTOP, KWTOP )
! 254: SI( KWTOP ) = ZERO
! 255: NS = 1
! 256: ND = 0
! 257: IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
! 258: $ THEN
! 259: NS = 0
! 260: ND = 1
! 261: IF( KWTOP.GT.KTOP )
! 262: $ H( KWTOP, KWTOP-1 ) = ZERO
! 263: END IF
! 264: WORK( 1 ) = ONE
! 265: RETURN
! 266: END IF
! 267: *
! 268: * ==== Convert to spike-triangular form. (In case of a
! 269: * . rare QR failure, this routine continues to do
! 270: * . aggressive early deflation using that part of
! 271: * . the deflation window that converged using INFQR
! 272: * . here and there to keep track.) ====
! 273: *
! 274: CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
! 275: CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
! 276: *
! 277: CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
! 278: NMIN = ILAENV( 12, 'DLAQR3', 'SV', JW, 1, JW, LWORK )
! 279: IF( JW.GT.NMIN ) THEN
! 280: CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
! 281: $ SI( KWTOP ), 1, JW, V, LDV, WORK, LWORK, INFQR )
! 282: ELSE
! 283: CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
! 284: $ SI( KWTOP ), 1, JW, V, LDV, INFQR )
! 285: END IF
! 286: *
! 287: * ==== DTREXC needs a clean margin near the diagonal ====
! 288: *
! 289: DO 10 J = 1, JW - 3
! 290: T( J+2, J ) = ZERO
! 291: T( J+3, J ) = ZERO
! 292: 10 CONTINUE
! 293: IF( JW.GT.2 )
! 294: $ T( JW, JW-2 ) = ZERO
! 295: *
! 296: * ==== Deflation detection loop ====
! 297: *
! 298: NS = JW
! 299: ILST = INFQR + 1
! 300: 20 CONTINUE
! 301: IF( ILST.LE.NS ) THEN
! 302: IF( NS.EQ.1 ) THEN
! 303: BULGE = .FALSE.
! 304: ELSE
! 305: BULGE = T( NS, NS-1 ).NE.ZERO
! 306: END IF
! 307: *
! 308: * ==== Small spike tip test for deflation ====
! 309: *
! 310: IF( .NOT.BULGE ) THEN
! 311: *
! 312: * ==== Real eigenvalue ====
! 313: *
! 314: FOO = ABS( T( NS, NS ) )
! 315: IF( FOO.EQ.ZERO )
! 316: $ FOO = ABS( S )
! 317: IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
! 318: *
! 319: * ==== Deflatable ====
! 320: *
! 321: NS = NS - 1
! 322: ELSE
! 323: *
! 324: * ==== Undeflatable. Move it up out of the way.
! 325: * . (DTREXC can not fail in this case.) ====
! 326: *
! 327: IFST = NS
! 328: CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
! 329: $ INFO )
! 330: ILST = ILST + 1
! 331: END IF
! 332: ELSE
! 333: *
! 334: * ==== Complex conjugate pair ====
! 335: *
! 336: FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
! 337: $ SQRT( ABS( T( NS-1, NS ) ) )
! 338: IF( FOO.EQ.ZERO )
! 339: $ FOO = ABS( S )
! 340: IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
! 341: $ MAX( SMLNUM, ULP*FOO ) ) THEN
! 342: *
! 343: * ==== Deflatable ====
! 344: *
! 345: NS = NS - 2
! 346: ELSE
! 347: *
! 348: * ==== Undeflatable. Move them up out of the way.
! 349: * . Fortunately, DTREXC does the right thing with
! 350: * . ILST in case of a rare exchange failure. ====
! 351: *
! 352: IFST = NS
! 353: CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
! 354: $ INFO )
! 355: ILST = ILST + 2
! 356: END IF
! 357: END IF
! 358: *
! 359: * ==== End deflation detection loop ====
! 360: *
! 361: GO TO 20
! 362: END IF
! 363: *
! 364: * ==== Return to Hessenberg form ====
! 365: *
! 366: IF( NS.EQ.0 )
! 367: $ S = ZERO
! 368: *
! 369: IF( NS.LT.JW ) THEN
! 370: *
! 371: * ==== sorting diagonal blocks of T improves accuracy for
! 372: * . graded matrices. Bubble sort deals well with
! 373: * . exchange failures. ====
! 374: *
! 375: SORTED = .false.
! 376: I = NS + 1
! 377: 30 CONTINUE
! 378: IF( SORTED )
! 379: $ GO TO 50
! 380: SORTED = .true.
! 381: *
! 382: KEND = I - 1
! 383: I = INFQR + 1
! 384: IF( I.EQ.NS ) THEN
! 385: K = I + 1
! 386: ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
! 387: K = I + 1
! 388: ELSE
! 389: K = I + 2
! 390: END IF
! 391: 40 CONTINUE
! 392: IF( K.LE.KEND ) THEN
! 393: IF( K.EQ.I+1 ) THEN
! 394: EVI = ABS( T( I, I ) )
! 395: ELSE
! 396: EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
! 397: $ SQRT( ABS( T( I, I+1 ) ) )
! 398: END IF
! 399: *
! 400: IF( K.EQ.KEND ) THEN
! 401: EVK = ABS( T( K, K ) )
! 402: ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
! 403: EVK = ABS( T( K, K ) )
! 404: ELSE
! 405: EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
! 406: $ SQRT( ABS( T( K, K+1 ) ) )
! 407: END IF
! 408: *
! 409: IF( EVI.GE.EVK ) THEN
! 410: I = K
! 411: ELSE
! 412: SORTED = .false.
! 413: IFST = I
! 414: ILST = K
! 415: CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
! 416: $ INFO )
! 417: IF( INFO.EQ.0 ) THEN
! 418: I = ILST
! 419: ELSE
! 420: I = K
! 421: END IF
! 422: END IF
! 423: IF( I.EQ.KEND ) THEN
! 424: K = I + 1
! 425: ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
! 426: K = I + 1
! 427: ELSE
! 428: K = I + 2
! 429: END IF
! 430: GO TO 40
! 431: END IF
! 432: GO TO 30
! 433: 50 CONTINUE
! 434: END IF
! 435: *
! 436: * ==== Restore shift/eigenvalue array from T ====
! 437: *
! 438: I = JW
! 439: 60 CONTINUE
! 440: IF( I.GE.INFQR+1 ) THEN
! 441: IF( I.EQ.INFQR+1 ) THEN
! 442: SR( KWTOP+I-1 ) = T( I, I )
! 443: SI( KWTOP+I-1 ) = ZERO
! 444: I = I - 1
! 445: ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
! 446: SR( KWTOP+I-1 ) = T( I, I )
! 447: SI( KWTOP+I-1 ) = ZERO
! 448: I = I - 1
! 449: ELSE
! 450: AA = T( I-1, I-1 )
! 451: CC = T( I, I-1 )
! 452: BB = T( I-1, I )
! 453: DD = T( I, I )
! 454: CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
! 455: $ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
! 456: $ SI( KWTOP+I-1 ), CS, SN )
! 457: I = I - 2
! 458: END IF
! 459: GO TO 60
! 460: END IF
! 461: *
! 462: IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
! 463: IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
! 464: *
! 465: * ==== Reflect spike back into lower triangle ====
! 466: *
! 467: CALL DCOPY( NS, V, LDV, WORK, 1 )
! 468: BETA = WORK( 1 )
! 469: CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
! 470: WORK( 1 ) = ONE
! 471: *
! 472: CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
! 473: *
! 474: CALL DLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
! 475: $ WORK( JW+1 ) )
! 476: CALL DLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
! 477: $ WORK( JW+1 ) )
! 478: CALL DLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
! 479: $ WORK( JW+1 ) )
! 480: *
! 481: CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
! 482: $ LWORK-JW, INFO )
! 483: END IF
! 484: *
! 485: * ==== Copy updated reduced window into place ====
! 486: *
! 487: IF( KWTOP.GT.1 )
! 488: $ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
! 489: CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
! 490: CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
! 491: $ LDH+1 )
! 492: *
! 493: * ==== Accumulate orthogonal matrix in order update
! 494: * . H and Z, if requested. ====
! 495: *
! 496: IF( NS.GT.1 .AND. S.NE.ZERO )
! 497: $ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
! 498: $ WORK( JW+1 ), LWORK-JW, INFO )
! 499: *
! 500: * ==== Update vertical slab in H ====
! 501: *
! 502: IF( WANTT ) THEN
! 503: LTOP = 1
! 504: ELSE
! 505: LTOP = KTOP
! 506: END IF
! 507: DO 70 KROW = LTOP, KWTOP - 1, NV
! 508: KLN = MIN( NV, KWTOP-KROW )
! 509: CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
! 510: $ LDH, V, LDV, ZERO, WV, LDWV )
! 511: CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
! 512: 70 CONTINUE
! 513: *
! 514: * ==== Update horizontal slab in H ====
! 515: *
! 516: IF( WANTT ) THEN
! 517: DO 80 KCOL = KBOT + 1, N, NH
! 518: KLN = MIN( NH, N-KCOL+1 )
! 519: CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
! 520: $ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
! 521: CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
! 522: $ LDH )
! 523: 80 CONTINUE
! 524: END IF
! 525: *
! 526: * ==== Update vertical slab in Z ====
! 527: *
! 528: IF( WANTZ ) THEN
! 529: DO 90 KROW = ILOZ, IHIZ, NV
! 530: KLN = MIN( NV, IHIZ-KROW+1 )
! 531: CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
! 532: $ LDZ, V, LDV, ZERO, WV, LDWV )
! 533: CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
! 534: $ LDZ )
! 535: 90 CONTINUE
! 536: END IF
! 537: END IF
! 538: *
! 539: * ==== Return the number of deflations ... ====
! 540: *
! 541: ND = JW - NS
! 542: *
! 543: * ==== ... and the number of shifts. (Subtracting
! 544: * . INFQR from the spike length takes care
! 545: * . of the case of a rare QR failure while
! 546: * . calculating eigenvalues of the deflation
! 547: * . window.) ====
! 548: *
! 549: NS = NS - INFQR
! 550: *
! 551: * ==== Return optimal workspace. ====
! 552: *
! 553: WORK( 1 ) = DBLE( LWKOPT )
! 554: *
! 555: * ==== End of DLAQR3 ====
! 556: *
! 557: END
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