Annotation of rpl/lapack/lapack/zlaqr3.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
! 2: $ IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
! 3: $ 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: COMPLEX*16 H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
! 16: $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
! 17: * ..
! 18: *
! 19: * ******************************************************************
! 20: * Aggressive early deflation:
! 21: *
! 22: * This subroutine accepts as input an upper Hessenberg matrix
! 23: * H and performs an unitary similarity transformation
! 24: * designed to detect and deflate fully converged eigenvalues from
! 25: * a trailing principal submatrix. On output H has been over-
! 26: * written by a new Hessenberg matrix that is a perturbation of
! 27: * an unitary similarity transformation of H. It is to be
! 28: * hoped that the final version of H has many zero subdiagonal
! 29: * entries.
! 30: *
! 31: * ******************************************************************
! 32: * WANTT (input) LOGICAL
! 33: * If .TRUE., then the Hessenberg matrix H is fully updated
! 34: * so that the triangular Schur factor may be
! 35: * computed (in cooperation with the calling subroutine).
! 36: * If .FALSE., then only enough of H is updated to preserve
! 37: * the eigenvalues.
! 38: *
! 39: * WANTZ (input) LOGICAL
! 40: * If .TRUE., then the unitary matrix Z is updated so
! 41: * so that the unitary Schur factor may be computed
! 42: * (in cooperation with the calling subroutine).
! 43: * If .FALSE., then Z is not referenced.
! 44: *
! 45: * N (input) INTEGER
! 46: * The order of the matrix H and (if WANTZ is .TRUE.) the
! 47: * order of the unitary matrix Z.
! 48: *
! 49: * KTOP (input) INTEGER
! 50: * It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
! 51: * KBOT and KTOP together determine an isolated block
! 52: * along the diagonal of the Hessenberg matrix.
! 53: *
! 54: * KBOT (input) INTEGER
! 55: * It is assumed without a check that either
! 56: * KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
! 57: * determine an isolated block along the diagonal of the
! 58: * Hessenberg matrix.
! 59: *
! 60: * NW (input) INTEGER
! 61: * Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
! 62: *
! 63: * H (input/output) COMPLEX*16 array, dimension (LDH,N)
! 64: * On input the initial N-by-N section of H stores the
! 65: * Hessenberg matrix undergoing aggressive early deflation.
! 66: * On output H has been transformed by a unitary
! 67: * similarity transformation, perturbed, and the returned
! 68: * to Hessenberg form that (it is to be hoped) has some
! 69: * zero subdiagonal entries.
! 70: *
! 71: * LDH (input) integer
! 72: * Leading dimension of H just as declared in the calling
! 73: * subroutine. N .LE. LDH
! 74: *
! 75: * ILOZ (input) INTEGER
! 76: * IHIZ (input) INTEGER
! 77: * Specify the rows of Z to which transformations must be
! 78: * applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
! 79: *
! 80: * Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
! 81: * IF WANTZ is .TRUE., then on output, the unitary
! 82: * similarity transformation mentioned above has been
! 83: * accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
! 84: * If WANTZ is .FALSE., then Z is unreferenced.
! 85: *
! 86: * LDZ (input) integer
! 87: * The leading dimension of Z just as declared in the
! 88: * calling subroutine. 1 .LE. LDZ.
! 89: *
! 90: * NS (output) integer
! 91: * The number of unconverged (ie approximate) eigenvalues
! 92: * returned in SR and SI that may be used as shifts by the
! 93: * calling subroutine.
! 94: *
! 95: * ND (output) integer
! 96: * The number of converged eigenvalues uncovered by this
! 97: * subroutine.
! 98: *
! 99: * SH (output) COMPLEX*16 array, dimension KBOT
! 100: * On output, approximate eigenvalues that may
! 101: * be used for shifts are stored in SH(KBOT-ND-NS+1)
! 102: * through SR(KBOT-ND). Converged eigenvalues are
! 103: * stored in SH(KBOT-ND+1) through SH(KBOT).
! 104: *
! 105: * V (workspace) COMPLEX*16 array, dimension (LDV,NW)
! 106: * An NW-by-NW work array.
! 107: *
! 108: * LDV (input) integer scalar
! 109: * The leading dimension of V just as declared in the
! 110: * calling subroutine. NW .LE. LDV
! 111: *
! 112: * NH (input) integer scalar
! 113: * The number of columns of T. NH.GE.NW.
! 114: *
! 115: * T (workspace) COMPLEX*16 array, dimension (LDT,NW)
! 116: *
! 117: * LDT (input) integer
! 118: * The leading dimension of T just as declared in the
! 119: * calling subroutine. NW .LE. LDT
! 120: *
! 121: * NV (input) integer
! 122: * The number of rows of work array WV available for
! 123: * workspace. NV.GE.NW.
! 124: *
! 125: * WV (workspace) COMPLEX*16 array, dimension (LDWV,NW)
! 126: *
! 127: * LDWV (input) integer
! 128: * The leading dimension of W just as declared in the
! 129: * calling subroutine. NW .LE. LDV
! 130: *
! 131: * WORK (workspace) COMPLEX*16 array, dimension LWORK.
! 132: * On exit, WORK(1) is set to an estimate of the optimal value
! 133: * of LWORK for the given values of N, NW, KTOP and KBOT.
! 134: *
! 135: * LWORK (input) integer
! 136: * The dimension of the work array WORK. LWORK = 2*NW
! 137: * suffices, but greater efficiency may result from larger
! 138: * values of LWORK.
! 139: *
! 140: * If LWORK = -1, then a workspace query is assumed; ZLAQR3
! 141: * only estimates the optimal workspace size for the given
! 142: * values of N, NW, KTOP and KBOT. The estimate is returned
! 143: * in WORK(1). No error message related to LWORK is issued
! 144: * by XERBLA. Neither H nor Z are accessed.
! 145: *
! 146: * ================================================================
! 147: * Based on contributions by
! 148: * Karen Braman and Ralph Byers, Department of Mathematics,
! 149: * University of Kansas, USA
! 150: *
! 151: * ================================================================
! 152: * .. Parameters ..
! 153: COMPLEX*16 ZERO, ONE
! 154: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
! 155: $ ONE = ( 1.0d0, 0.0d0 ) )
! 156: DOUBLE PRECISION RZERO, RONE
! 157: PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0 )
! 158: * ..
! 159: * .. Local Scalars ..
! 160: COMPLEX*16 BETA, CDUM, S, TAU
! 161: DOUBLE PRECISION FOO, SAFMAX, SAFMIN, SMLNUM, ULP
! 162: INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
! 163: $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
! 164: $ LWKOPT, NMIN
! 165: * ..
! 166: * .. External Functions ..
! 167: DOUBLE PRECISION DLAMCH
! 168: INTEGER ILAENV
! 169: EXTERNAL DLAMCH, ILAENV
! 170: * ..
! 171: * .. External Subroutines ..
! 172: EXTERNAL DLABAD, ZCOPY, ZGEHRD, ZGEMM, ZLACPY, ZLAHQR,
! 173: $ ZLAQR4, ZLARF, ZLARFG, ZLASET, ZTREXC, ZUNMHR
! 174: * ..
! 175: * .. Intrinsic Functions ..
! 176: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, MAX, MIN
! 177: * ..
! 178: * .. Statement Functions ..
! 179: DOUBLE PRECISION CABS1
! 180: * ..
! 181: * .. Statement Function definitions ..
! 182: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
! 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 ZGEHRD ====
! 194: *
! 195: CALL ZGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
! 196: LWK1 = INT( WORK( 1 ) )
! 197: *
! 198: * ==== Workspace query call to ZUNMHR ====
! 199: *
! 200: CALL ZUNMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
! 201: $ WORK, -1, INFO )
! 202: LWK2 = INT( WORK( 1 ) )
! 203: *
! 204: * ==== Workspace query call to ZLAQR4 ====
! 205: *
! 206: CALL ZLAQR4( .true., .true., JW, 1, JW, T, LDT, SH, 1, JW, V,
! 207: $ LDV, WORK, -1, INFQR )
! 208: LWK3 = INT( WORK( 1 ) )
! 209: *
! 210: * ==== Optimal workspace ====
! 211: *
! 212: LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
! 213: END IF
! 214: *
! 215: * ==== Quick return in case of workspace query. ====
! 216: *
! 217: IF( LWORK.EQ.-1 ) THEN
! 218: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
! 219: RETURN
! 220: END IF
! 221: *
! 222: * ==== Nothing to do ...
! 223: * ... for an empty active block ... ====
! 224: NS = 0
! 225: ND = 0
! 226: WORK( 1 ) = ONE
! 227: IF( KTOP.GT.KBOT )
! 228: $ RETURN
! 229: * ... nor for an empty deflation window. ====
! 230: IF( NW.LT.1 )
! 231: $ RETURN
! 232: *
! 233: * ==== Machine constants ====
! 234: *
! 235: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
! 236: SAFMAX = RONE / SAFMIN
! 237: CALL DLABAD( SAFMIN, SAFMAX )
! 238: ULP = DLAMCH( 'PRECISION' )
! 239: SMLNUM = SAFMIN*( DBLE( N ) / ULP )
! 240: *
! 241: * ==== Setup deflation window ====
! 242: *
! 243: JW = MIN( NW, KBOT-KTOP+1 )
! 244: KWTOP = KBOT - JW + 1
! 245: IF( KWTOP.EQ.KTOP ) THEN
! 246: S = ZERO
! 247: ELSE
! 248: S = H( KWTOP, KWTOP-1 )
! 249: END IF
! 250: *
! 251: IF( KBOT.EQ.KWTOP ) THEN
! 252: *
! 253: * ==== 1-by-1 deflation window: not much to do ====
! 254: *
! 255: SH( KWTOP ) = H( KWTOP, KWTOP )
! 256: NS = 1
! 257: ND = 0
! 258: IF( CABS1( S ).LE.MAX( SMLNUM, ULP*CABS1( H( KWTOP,
! 259: $ KWTOP ) ) ) ) THEN
! 260: NS = 0
! 261: ND = 1
! 262: IF( KWTOP.GT.KTOP )
! 263: $ H( KWTOP, KWTOP-1 ) = ZERO
! 264: END IF
! 265: WORK( 1 ) = ONE
! 266: RETURN
! 267: END IF
! 268: *
! 269: * ==== Convert to spike-triangular form. (In case of a
! 270: * . rare QR failure, this routine continues to do
! 271: * . aggressive early deflation using that part of
! 272: * . the deflation window that converged using INFQR
! 273: * . here and there to keep track.) ====
! 274: *
! 275: CALL ZLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
! 276: CALL ZCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
! 277: *
! 278: CALL ZLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
! 279: NMIN = ILAENV( 12, 'ZLAQR3', 'SV', JW, 1, JW, LWORK )
! 280: IF( JW.GT.NMIN ) THEN
! 281: CALL ZLAQR4( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
! 282: $ JW, V, LDV, WORK, LWORK, INFQR )
! 283: ELSE
! 284: CALL ZLAHQR( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
! 285: $ JW, V, LDV, INFQR )
! 286: END IF
! 287: *
! 288: * ==== Deflation detection loop ====
! 289: *
! 290: NS = JW
! 291: ILST = INFQR + 1
! 292: DO 10 KNT = INFQR + 1, JW
! 293: *
! 294: * ==== Small spike tip deflation test ====
! 295: *
! 296: FOO = CABS1( T( NS, NS ) )
! 297: IF( FOO.EQ.RZERO )
! 298: $ FOO = CABS1( S )
! 299: IF( CABS1( S )*CABS1( V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) )
! 300: $ THEN
! 301: *
! 302: * ==== One more converged eigenvalue ====
! 303: *
! 304: NS = NS - 1
! 305: ELSE
! 306: *
! 307: * ==== One undeflatable eigenvalue. Move it up out of the
! 308: * . way. (ZTREXC can not fail in this case.) ====
! 309: *
! 310: IFST = NS
! 311: CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
! 312: ILST = ILST + 1
! 313: END IF
! 314: 10 CONTINUE
! 315: *
! 316: * ==== Return to Hessenberg form ====
! 317: *
! 318: IF( NS.EQ.0 )
! 319: $ S = ZERO
! 320: *
! 321: IF( NS.LT.JW ) THEN
! 322: *
! 323: * ==== sorting the diagonal of T improves accuracy for
! 324: * . graded matrices. ====
! 325: *
! 326: DO 30 I = INFQR + 1, NS
! 327: IFST = I
! 328: DO 20 J = I + 1, NS
! 329: IF( CABS1( T( J, J ) ).GT.CABS1( T( IFST, IFST ) ) )
! 330: $ IFST = J
! 331: 20 CONTINUE
! 332: ILST = I
! 333: IF( IFST.NE.ILST )
! 334: $ CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
! 335: 30 CONTINUE
! 336: END IF
! 337: *
! 338: * ==== Restore shift/eigenvalue array from T ====
! 339: *
! 340: DO 40 I = INFQR + 1, JW
! 341: SH( KWTOP+I-1 ) = T( I, I )
! 342: 40 CONTINUE
! 343: *
! 344: *
! 345: IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
! 346: IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
! 347: *
! 348: * ==== Reflect spike back into lower triangle ====
! 349: *
! 350: CALL ZCOPY( NS, V, LDV, WORK, 1 )
! 351: DO 50 I = 1, NS
! 352: WORK( I ) = DCONJG( WORK( I ) )
! 353: 50 CONTINUE
! 354: BETA = WORK( 1 )
! 355: CALL ZLARFG( NS, BETA, WORK( 2 ), 1, TAU )
! 356: WORK( 1 ) = ONE
! 357: *
! 358: CALL ZLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
! 359: *
! 360: CALL ZLARF( 'L', NS, JW, WORK, 1, DCONJG( TAU ), T, LDT,
! 361: $ WORK( JW+1 ) )
! 362: CALL ZLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
! 363: $ WORK( JW+1 ) )
! 364: CALL ZLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
! 365: $ WORK( JW+1 ) )
! 366: *
! 367: CALL ZGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
! 368: $ LWORK-JW, INFO )
! 369: END IF
! 370: *
! 371: * ==== Copy updated reduced window into place ====
! 372: *
! 373: IF( KWTOP.GT.1 )
! 374: $ H( KWTOP, KWTOP-1 ) = S*DCONJG( V( 1, 1 ) )
! 375: CALL ZLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
! 376: CALL ZCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
! 377: $ LDH+1 )
! 378: *
! 379: * ==== Accumulate orthogonal matrix in order update
! 380: * . H and Z, if requested. ====
! 381: *
! 382: IF( NS.GT.1 .AND. S.NE.ZERO )
! 383: $ CALL ZUNMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
! 384: $ WORK( JW+1 ), LWORK-JW, INFO )
! 385: *
! 386: * ==== Update vertical slab in H ====
! 387: *
! 388: IF( WANTT ) THEN
! 389: LTOP = 1
! 390: ELSE
! 391: LTOP = KTOP
! 392: END IF
! 393: DO 60 KROW = LTOP, KWTOP - 1, NV
! 394: KLN = MIN( NV, KWTOP-KROW )
! 395: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
! 396: $ LDH, V, LDV, ZERO, WV, LDWV )
! 397: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
! 398: 60 CONTINUE
! 399: *
! 400: * ==== Update horizontal slab in H ====
! 401: *
! 402: IF( WANTT ) THEN
! 403: DO 70 KCOL = KBOT + 1, N, NH
! 404: KLN = MIN( NH, N-KCOL+1 )
! 405: CALL ZGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
! 406: $ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
! 407: CALL ZLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
! 408: $ LDH )
! 409: 70 CONTINUE
! 410: END IF
! 411: *
! 412: * ==== Update vertical slab in Z ====
! 413: *
! 414: IF( WANTZ ) THEN
! 415: DO 80 KROW = ILOZ, IHIZ, NV
! 416: KLN = MIN( NV, IHIZ-KROW+1 )
! 417: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
! 418: $ LDZ, V, LDV, ZERO, WV, LDWV )
! 419: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
! 420: $ LDZ )
! 421: 80 CONTINUE
! 422: END IF
! 423: END IF
! 424: *
! 425: * ==== Return the number of deflations ... ====
! 426: *
! 427: ND = JW - NS
! 428: *
! 429: * ==== ... and the number of shifts. (Subtracting
! 430: * . INFQR from the spike length takes care
! 431: * . of the case of a rare QR failure while
! 432: * . calculating eigenvalues of the deflation
! 433: * . window.) ====
! 434: *
! 435: NS = NS - INFQR
! 436: *
! 437: * ==== Return optimal workspace. ====
! 438: *
! 439: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
! 440: *
! 441: * ==== End of ZLAQR3 ====
! 442: *
! 443: END
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