Annotation of rpl/lapack/lapack/dlaqr2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLAQR2( 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: * This subroutine is identical to DLAQR3 except that it avoids
! 21: * recursion by calling DLAHQR instead of DLAQR4.
! 22: *
! 23: *
! 24: * ******************************************************************
! 25: * Aggressive early deflation:
! 26: *
! 27: * This subroutine accepts as input an upper Hessenberg matrix
! 28: * H and performs an orthogonal similarity transformation
! 29: * designed to detect and deflate fully converged eigenvalues from
! 30: * a trailing principal submatrix. On output H has been over-
! 31: * written by a new Hessenberg matrix that is a perturbation of
! 32: * an orthogonal similarity transformation of H. It is to be
! 33: * hoped that the final version of H has many zero subdiagonal
! 34: * entries.
! 35: *
! 36: * ******************************************************************
! 37: * WANTT (input) LOGICAL
! 38: * If .TRUE., then the Hessenberg matrix H is fully updated
! 39: * so that the quasi-triangular Schur factor may be
! 40: * computed (in cooperation with the calling subroutine).
! 41: * If .FALSE., then only enough of H is updated to preserve
! 42: * the eigenvalues.
! 43: *
! 44: * WANTZ (input) LOGICAL
! 45: * If .TRUE., then the orthogonal matrix Z is updated so
! 46: * so that the orthogonal Schur factor may be computed
! 47: * (in cooperation with the calling subroutine).
! 48: * If .FALSE., then Z is not referenced.
! 49: *
! 50: * N (input) INTEGER
! 51: * The order of the matrix H and (if WANTZ is .TRUE.) the
! 52: * order of the orthogonal matrix Z.
! 53: *
! 54: * KTOP (input) INTEGER
! 55: * It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
! 56: * KBOT and KTOP together determine an isolated block
! 57: * along the diagonal of the Hessenberg matrix.
! 58: *
! 59: * KBOT (input) INTEGER
! 60: * It is assumed without a check that either
! 61: * KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
! 62: * determine an isolated block along the diagonal of the
! 63: * Hessenberg matrix.
! 64: *
! 65: * NW (input) INTEGER
! 66: * Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
! 67: *
! 68: * H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
! 69: * On input the initial N-by-N section of H stores the
! 70: * Hessenberg matrix undergoing aggressive early deflation.
! 71: * On output H has been transformed by an orthogonal
! 72: * similarity transformation, perturbed, and the returned
! 73: * to Hessenberg form that (it is to be hoped) has some
! 74: * zero subdiagonal entries.
! 75: *
! 76: * LDH (input) integer
! 77: * Leading dimension of H just as declared in the calling
! 78: * subroutine. N .LE. LDH
! 79: *
! 80: * ILOZ (input) INTEGER
! 81: * IHIZ (input) INTEGER
! 82: * Specify the rows of Z to which transformations must be
! 83: * applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
! 84: *
! 85: * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
! 86: * IF WANTZ is .TRUE., then on output, the orthogonal
! 87: * similarity transformation mentioned above has been
! 88: * accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
! 89: * If WANTZ is .FALSE., then Z is unreferenced.
! 90: *
! 91: * LDZ (input) integer
! 92: * The leading dimension of Z just as declared in the
! 93: * calling subroutine. 1 .LE. LDZ.
! 94: *
! 95: * NS (output) integer
! 96: * The number of unconverged (ie approximate) eigenvalues
! 97: * returned in SR and SI that may be used as shifts by the
! 98: * calling subroutine.
! 99: *
! 100: * ND (output) integer
! 101: * The number of converged eigenvalues uncovered by this
! 102: * subroutine.
! 103: *
! 104: * SR (output) DOUBLE PRECISION array, dimension KBOT
! 105: * SI (output) DOUBLE PRECISION array, dimension KBOT
! 106: * On output, the real and imaginary parts of approximate
! 107: * eigenvalues that may be used for shifts are stored in
! 108: * SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
! 109: * SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
! 110: * The real and imaginary parts of converged eigenvalues
! 111: * are stored in SR(KBOT-ND+1) through SR(KBOT) and
! 112: * SI(KBOT-ND+1) through SI(KBOT), respectively.
! 113: *
! 114: * V (workspace) DOUBLE PRECISION array, dimension (LDV,NW)
! 115: * An NW-by-NW work array.
! 116: *
! 117: * LDV (input) integer scalar
! 118: * The leading dimension of V just as declared in the
! 119: * calling subroutine. NW .LE. LDV
! 120: *
! 121: * NH (input) integer scalar
! 122: * The number of columns of T. NH.GE.NW.
! 123: *
! 124: * T (workspace) DOUBLE PRECISION array, dimension (LDT,NW)
! 125: *
! 126: * LDT (input) integer
! 127: * The leading dimension of T just as declared in the
! 128: * calling subroutine. NW .LE. LDT
! 129: *
! 130: * NV (input) integer
! 131: * The number of rows of work array WV available for
! 132: * workspace. NV.GE.NW.
! 133: *
! 134: * WV (workspace) DOUBLE PRECISION array, dimension (LDWV,NW)
! 135: *
! 136: * LDWV (input) integer
! 137: * The leading dimension of W just as declared in the
! 138: * calling subroutine. NW .LE. LDV
! 139: *
! 140: * WORK (workspace) DOUBLE PRECISION array, dimension LWORK.
! 141: * On exit, WORK(1) is set to an estimate of the optimal value
! 142: * of LWORK for the given values of N, NW, KTOP and KBOT.
! 143: *
! 144: * LWORK (input) integer
! 145: * The dimension of the work array WORK. LWORK = 2*NW
! 146: * suffices, but greater efficiency may result from larger
! 147: * values of LWORK.
! 148: *
! 149: * If LWORK = -1, then a workspace query is assumed; DLAQR2
! 150: * only estimates the optimal workspace size for the given
! 151: * values of N, NW, KTOP and KBOT. The estimate is returned
! 152: * in WORK(1). No error message related to LWORK is issued
! 153: * by XERBLA. Neither H nor Z are accessed.
! 154: *
! 155: * ================================================================
! 156: * Based on contributions by
! 157: * Karen Braman and Ralph Byers, Department of Mathematics,
! 158: * University of Kansas, USA
! 159: *
! 160: * ================================================================
! 161: * .. Parameters ..
! 162: DOUBLE PRECISION ZERO, ONE
! 163: PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
! 164: * ..
! 165: * .. Local Scalars ..
! 166: DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
! 167: $ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
! 168: INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
! 169: $ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2,
! 170: $ LWKOPT
! 171: LOGICAL BULGE, SORTED
! 172: * ..
! 173: * .. External Functions ..
! 174: DOUBLE PRECISION DLAMCH
! 175: EXTERNAL DLAMCH
! 176: * ..
! 177: * .. External Subroutines ..
! 178: EXTERNAL DCOPY, DGEHRD, DGEMM, DLABAD, DLACPY, DLAHQR,
! 179: $ DLANV2, DLARF, DLARFG, DLASET, DORMHR, DTREXC
! 180: * ..
! 181: * .. Intrinsic Functions ..
! 182: INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
! 183: * ..
! 184: * .. Executable Statements ..
! 185: *
! 186: * ==== Estimate optimal workspace. ====
! 187: *
! 188: JW = MIN( NW, KBOT-KTOP+1 )
! 189: IF( JW.LE.2 ) THEN
! 190: LWKOPT = 1
! 191: ELSE
! 192: *
! 193: * ==== Workspace query call to DGEHRD ====
! 194: *
! 195: CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
! 196: LWK1 = INT( WORK( 1 ) )
! 197: *
! 198: * ==== Workspace query call to DORMHR ====
! 199: *
! 200: CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
! 201: $ WORK, -1, INFO )
! 202: LWK2 = INT( WORK( 1 ) )
! 203: *
! 204: * ==== Optimal workspace ====
! 205: *
! 206: LWKOPT = JW + MAX( LWK1, LWK2 )
! 207: END IF
! 208: *
! 209: * ==== Quick return in case of workspace query. ====
! 210: *
! 211: IF( LWORK.EQ.-1 ) THEN
! 212: WORK( 1 ) = DBLE( LWKOPT )
! 213: RETURN
! 214: END IF
! 215: *
! 216: * ==== Nothing to do ...
! 217: * ... for an empty active block ... ====
! 218: NS = 0
! 219: ND = 0
! 220: WORK( 1 ) = ONE
! 221: IF( KTOP.GT.KBOT )
! 222: $ RETURN
! 223: * ... nor for an empty deflation window. ====
! 224: IF( NW.LT.1 )
! 225: $ RETURN
! 226: *
! 227: * ==== Machine constants ====
! 228: *
! 229: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
! 230: SAFMAX = ONE / SAFMIN
! 231: CALL DLABAD( SAFMIN, SAFMAX )
! 232: ULP = DLAMCH( 'PRECISION' )
! 233: SMLNUM = SAFMIN*( DBLE( N ) / ULP )
! 234: *
! 235: * ==== Setup deflation window ====
! 236: *
! 237: JW = MIN( NW, KBOT-KTOP+1 )
! 238: KWTOP = KBOT - JW + 1
! 239: IF( KWTOP.EQ.KTOP ) THEN
! 240: S = ZERO
! 241: ELSE
! 242: S = H( KWTOP, KWTOP-1 )
! 243: END IF
! 244: *
! 245: IF( KBOT.EQ.KWTOP ) THEN
! 246: *
! 247: * ==== 1-by-1 deflation window: not much to do ====
! 248: *
! 249: SR( KWTOP ) = H( KWTOP, KWTOP )
! 250: SI( KWTOP ) = ZERO
! 251: NS = 1
! 252: ND = 0
! 253: IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
! 254: $ THEN
! 255: NS = 0
! 256: ND = 1
! 257: IF( KWTOP.GT.KTOP )
! 258: $ H( KWTOP, KWTOP-1 ) = ZERO
! 259: END IF
! 260: WORK( 1 ) = ONE
! 261: RETURN
! 262: END IF
! 263: *
! 264: * ==== Convert to spike-triangular form. (In case of a
! 265: * . rare QR failure, this routine continues to do
! 266: * . aggressive early deflation using that part of
! 267: * . the deflation window that converged using INFQR
! 268: * . here and there to keep track.) ====
! 269: *
! 270: CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
! 271: CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
! 272: *
! 273: CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
! 274: CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
! 275: $ SI( KWTOP ), 1, JW, V, LDV, INFQR )
! 276: *
! 277: * ==== DTREXC needs a clean margin near the diagonal ====
! 278: *
! 279: DO 10 J = 1, JW - 3
! 280: T( J+2, J ) = ZERO
! 281: T( J+3, J ) = ZERO
! 282: 10 CONTINUE
! 283: IF( JW.GT.2 )
! 284: $ T( JW, JW-2 ) = ZERO
! 285: *
! 286: * ==== Deflation detection loop ====
! 287: *
! 288: NS = JW
! 289: ILST = INFQR + 1
! 290: 20 CONTINUE
! 291: IF( ILST.LE.NS ) THEN
! 292: IF( NS.EQ.1 ) THEN
! 293: BULGE = .FALSE.
! 294: ELSE
! 295: BULGE = T( NS, NS-1 ).NE.ZERO
! 296: END IF
! 297: *
! 298: * ==== Small spike tip test for deflation ====
! 299: *
! 300: IF( .NOT.BULGE ) THEN
! 301: *
! 302: * ==== Real eigenvalue ====
! 303: *
! 304: FOO = ABS( T( NS, NS ) )
! 305: IF( FOO.EQ.ZERO )
! 306: $ FOO = ABS( S )
! 307: IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
! 308: *
! 309: * ==== Deflatable ====
! 310: *
! 311: NS = NS - 1
! 312: ELSE
! 313: *
! 314: * ==== Undeflatable. Move it up out of the way.
! 315: * . (DTREXC can not fail in this case.) ====
! 316: *
! 317: IFST = NS
! 318: CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
! 319: $ INFO )
! 320: ILST = ILST + 1
! 321: END IF
! 322: ELSE
! 323: *
! 324: * ==== Complex conjugate pair ====
! 325: *
! 326: FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
! 327: $ SQRT( ABS( T( NS-1, NS ) ) )
! 328: IF( FOO.EQ.ZERO )
! 329: $ FOO = ABS( S )
! 330: IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
! 331: $ MAX( SMLNUM, ULP*FOO ) ) THEN
! 332: *
! 333: * ==== Deflatable ====
! 334: *
! 335: NS = NS - 2
! 336: ELSE
! 337: *
! 338: * ==== Undeflatable. Move them up out of the way.
! 339: * . Fortunately, DTREXC does the right thing with
! 340: * . ILST in case of a rare exchange failure. ====
! 341: *
! 342: IFST = NS
! 343: CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
! 344: $ INFO )
! 345: ILST = ILST + 2
! 346: END IF
! 347: END IF
! 348: *
! 349: * ==== End deflation detection loop ====
! 350: *
! 351: GO TO 20
! 352: END IF
! 353: *
! 354: * ==== Return to Hessenberg form ====
! 355: *
! 356: IF( NS.EQ.0 )
! 357: $ S = ZERO
! 358: *
! 359: IF( NS.LT.JW ) THEN
! 360: *
! 361: * ==== sorting diagonal blocks of T improves accuracy for
! 362: * . graded matrices. Bubble sort deals well with
! 363: * . exchange failures. ====
! 364: *
! 365: SORTED = .false.
! 366: I = NS + 1
! 367: 30 CONTINUE
! 368: IF( SORTED )
! 369: $ GO TO 50
! 370: SORTED = .true.
! 371: *
! 372: KEND = I - 1
! 373: I = INFQR + 1
! 374: IF( I.EQ.NS ) THEN
! 375: K = I + 1
! 376: ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
! 377: K = I + 1
! 378: ELSE
! 379: K = I + 2
! 380: END IF
! 381: 40 CONTINUE
! 382: IF( K.LE.KEND ) THEN
! 383: IF( K.EQ.I+1 ) THEN
! 384: EVI = ABS( T( I, I ) )
! 385: ELSE
! 386: EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
! 387: $ SQRT( ABS( T( I, I+1 ) ) )
! 388: END IF
! 389: *
! 390: IF( K.EQ.KEND ) THEN
! 391: EVK = ABS( T( K, K ) )
! 392: ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
! 393: EVK = ABS( T( K, K ) )
! 394: ELSE
! 395: EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
! 396: $ SQRT( ABS( T( K, K+1 ) ) )
! 397: END IF
! 398: *
! 399: IF( EVI.GE.EVK ) THEN
! 400: I = K
! 401: ELSE
! 402: SORTED = .false.
! 403: IFST = I
! 404: ILST = K
! 405: CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
! 406: $ INFO )
! 407: IF( INFO.EQ.0 ) THEN
! 408: I = ILST
! 409: ELSE
! 410: I = K
! 411: END IF
! 412: END IF
! 413: IF( I.EQ.KEND ) THEN
! 414: K = I + 1
! 415: ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
! 416: K = I + 1
! 417: ELSE
! 418: K = I + 2
! 419: END IF
! 420: GO TO 40
! 421: END IF
! 422: GO TO 30
! 423: 50 CONTINUE
! 424: END IF
! 425: *
! 426: * ==== Restore shift/eigenvalue array from T ====
! 427: *
! 428: I = JW
! 429: 60 CONTINUE
! 430: IF( I.GE.INFQR+1 ) THEN
! 431: IF( I.EQ.INFQR+1 ) THEN
! 432: SR( KWTOP+I-1 ) = T( I, I )
! 433: SI( KWTOP+I-1 ) = ZERO
! 434: I = I - 1
! 435: ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
! 436: SR( KWTOP+I-1 ) = T( I, I )
! 437: SI( KWTOP+I-1 ) = ZERO
! 438: I = I - 1
! 439: ELSE
! 440: AA = T( I-1, I-1 )
! 441: CC = T( I, I-1 )
! 442: BB = T( I-1, I )
! 443: DD = T( I, I )
! 444: CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
! 445: $ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
! 446: $ SI( KWTOP+I-1 ), CS, SN )
! 447: I = I - 2
! 448: END IF
! 449: GO TO 60
! 450: END IF
! 451: *
! 452: IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
! 453: IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
! 454: *
! 455: * ==== Reflect spike back into lower triangle ====
! 456: *
! 457: CALL DCOPY( NS, V, LDV, WORK, 1 )
! 458: BETA = WORK( 1 )
! 459: CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
! 460: WORK( 1 ) = ONE
! 461: *
! 462: CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
! 463: *
! 464: CALL DLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
! 465: $ WORK( JW+1 ) )
! 466: CALL DLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
! 467: $ WORK( JW+1 ) )
! 468: CALL DLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
! 469: $ WORK( JW+1 ) )
! 470: *
! 471: CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
! 472: $ LWORK-JW, INFO )
! 473: END IF
! 474: *
! 475: * ==== Copy updated reduced window into place ====
! 476: *
! 477: IF( KWTOP.GT.1 )
! 478: $ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
! 479: CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
! 480: CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
! 481: $ LDH+1 )
! 482: *
! 483: * ==== Accumulate orthogonal matrix in order update
! 484: * . H and Z, if requested. ====
! 485: *
! 486: IF( NS.GT.1 .AND. S.NE.ZERO )
! 487: $ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
! 488: $ WORK( JW+1 ), LWORK-JW, INFO )
! 489: *
! 490: * ==== Update vertical slab in H ====
! 491: *
! 492: IF( WANTT ) THEN
! 493: LTOP = 1
! 494: ELSE
! 495: LTOP = KTOP
! 496: END IF
! 497: DO 70 KROW = LTOP, KWTOP - 1, NV
! 498: KLN = MIN( NV, KWTOP-KROW )
! 499: CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
! 500: $ LDH, V, LDV, ZERO, WV, LDWV )
! 501: CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
! 502: 70 CONTINUE
! 503: *
! 504: * ==== Update horizontal slab in H ====
! 505: *
! 506: IF( WANTT ) THEN
! 507: DO 80 KCOL = KBOT + 1, N, NH
! 508: KLN = MIN( NH, N-KCOL+1 )
! 509: CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
! 510: $ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
! 511: CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
! 512: $ LDH )
! 513: 80 CONTINUE
! 514: END IF
! 515: *
! 516: * ==== Update vertical slab in Z ====
! 517: *
! 518: IF( WANTZ ) THEN
! 519: DO 90 KROW = ILOZ, IHIZ, NV
! 520: KLN = MIN( NV, IHIZ-KROW+1 )
! 521: CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
! 522: $ LDZ, V, LDV, ZERO, WV, LDWV )
! 523: CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
! 524: $ LDZ )
! 525: 90 CONTINUE
! 526: END IF
! 527: END IF
! 528: *
! 529: * ==== Return the number of deflations ... ====
! 530: *
! 531: ND = JW - NS
! 532: *
! 533: * ==== ... and the number of shifts. (Subtracting
! 534: * . INFQR from the spike length takes care
! 535: * . of the case of a rare QR failure while
! 536: * . calculating eigenvalues of the deflation
! 537: * . window.) ====
! 538: *
! 539: NS = NS - INFQR
! 540: *
! 541: * ==== Return optimal workspace. ====
! 542: *
! 543: WORK( 1 ) = DBLE( LWKOPT )
! 544: *
! 545: * ==== End of DLAQR2 ====
! 546: *
! 547: END
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