Annotation of rpl/lapack/lapack/zlaqr2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAQR2( 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: * This subroutine is identical to ZLAQR3 except that it avoids
! 20: * recursion by calling ZLAHQR instead of ZLAQR4.
! 21: *
! 22: *
! 23: * ******************************************************************
! 24: * Aggressive early deflation:
! 25: *
! 26: * This subroutine accepts as input an upper Hessenberg matrix
! 27: * H and performs an unitary similarity transformation
! 28: * designed to detect and deflate fully converged eigenvalues from
! 29: * a trailing principal submatrix. On output H has been over-
! 30: * written by a new Hessenberg matrix that is a perturbation of
! 31: * an unitary similarity transformation of H. It is to be
! 32: * hoped that the final version of H has many zero subdiagonal
! 33: * entries.
! 34: *
! 35: * ******************************************************************
! 36: * WANTT (input) LOGICAL
! 37: * If .TRUE., then the Hessenberg matrix H is fully updated
! 38: * so that the triangular Schur factor may be
! 39: * computed (in cooperation with the calling subroutine).
! 40: * If .FALSE., then only enough of H is updated to preserve
! 41: * the eigenvalues.
! 42: *
! 43: * WANTZ (input) LOGICAL
! 44: * If .TRUE., then the unitary matrix Z is updated so
! 45: * so that the unitary Schur factor may be computed
! 46: * (in cooperation with the calling subroutine).
! 47: * If .FALSE., then Z is not referenced.
! 48: *
! 49: * N (input) INTEGER
! 50: * The order of the matrix H and (if WANTZ is .TRUE.) the
! 51: * order of the unitary matrix Z.
! 52: *
! 53: * KTOP (input) INTEGER
! 54: * It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
! 55: * KBOT and KTOP together determine an isolated block
! 56: * along the diagonal of the Hessenberg matrix.
! 57: *
! 58: * KBOT (input) INTEGER
! 59: * It is assumed without a check that either
! 60: * KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
! 61: * determine an isolated block along the diagonal of the
! 62: * Hessenberg matrix.
! 63: *
! 64: * NW (input) INTEGER
! 65: * Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
! 66: *
! 67: * H (input/output) COMPLEX*16 array, dimension (LDH,N)
! 68: * On input the initial N-by-N section of H stores the
! 69: * Hessenberg matrix undergoing aggressive early deflation.
! 70: * On output H has been transformed by a unitary
! 71: * similarity transformation, perturbed, and the returned
! 72: * to Hessenberg form that (it is to be hoped) has some
! 73: * zero subdiagonal entries.
! 74: *
! 75: * LDH (input) integer
! 76: * Leading dimension of H just as declared in the calling
! 77: * subroutine. N .LE. LDH
! 78: *
! 79: * ILOZ (input) INTEGER
! 80: * IHIZ (input) INTEGER
! 81: * Specify the rows of Z to which transformations must be
! 82: * applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
! 83: *
! 84: * Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
! 85: * IF WANTZ is .TRUE., then on output, the unitary
! 86: * similarity transformation mentioned above has been
! 87: * accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
! 88: * If WANTZ is .FALSE., then Z is unreferenced.
! 89: *
! 90: * LDZ (input) integer
! 91: * The leading dimension of Z just as declared in the
! 92: * calling subroutine. 1 .LE. LDZ.
! 93: *
! 94: * NS (output) integer
! 95: * The number of unconverged (ie approximate) eigenvalues
! 96: * returned in SR and SI that may be used as shifts by the
! 97: * calling subroutine.
! 98: *
! 99: * ND (output) integer
! 100: * The number of converged eigenvalues uncovered by this
! 101: * subroutine.
! 102: *
! 103: * SH (output) COMPLEX*16 array, dimension KBOT
! 104: * On output, approximate eigenvalues that may
! 105: * be used for shifts are stored in SH(KBOT-ND-NS+1)
! 106: * through SR(KBOT-ND). Converged eigenvalues are
! 107: * stored in SH(KBOT-ND+1) through SH(KBOT).
! 108: *
! 109: * V (workspace) COMPLEX*16 array, dimension (LDV,NW)
! 110: * An NW-by-NW work array.
! 111: *
! 112: * LDV (input) integer scalar
! 113: * The leading dimension of V just as declared in the
! 114: * calling subroutine. NW .LE. LDV
! 115: *
! 116: * NH (input) integer scalar
! 117: * The number of columns of T. NH.GE.NW.
! 118: *
! 119: * T (workspace) COMPLEX*16 array, dimension (LDT,NW)
! 120: *
! 121: * LDT (input) integer
! 122: * The leading dimension of T just as declared in the
! 123: * calling subroutine. NW .LE. LDT
! 124: *
! 125: * NV (input) integer
! 126: * The number of rows of work array WV available for
! 127: * workspace. NV.GE.NW.
! 128: *
! 129: * WV (workspace) COMPLEX*16 array, dimension (LDWV,NW)
! 130: *
! 131: * LDWV (input) integer
! 132: * The leading dimension of W just as declared in the
! 133: * calling subroutine. NW .LE. LDV
! 134: *
! 135: * WORK (workspace) COMPLEX*16 array, dimension LWORK.
! 136: * On exit, WORK(1) is set to an estimate of the optimal value
! 137: * of LWORK for the given values of N, NW, KTOP and KBOT.
! 138: *
! 139: * LWORK (input) integer
! 140: * The dimension of the work array WORK. LWORK = 2*NW
! 141: * suffices, but greater efficiency may result from larger
! 142: * values of LWORK.
! 143: *
! 144: * If LWORK = -1, then a workspace query is assumed; ZLAQR2
! 145: * only estimates the optimal workspace size for the given
! 146: * values of N, NW, KTOP and KBOT. The estimate is returned
! 147: * in WORK(1). No error message related to LWORK is issued
! 148: * by XERBLA. Neither H nor Z are accessed.
! 149: *
! 150: * ================================================================
! 151: * Based on contributions by
! 152: * Karen Braman and Ralph Byers, Department of Mathematics,
! 153: * University of Kansas, USA
! 154: *
! 155: * ================================================================
! 156: * .. Parameters ..
! 157: COMPLEX*16 ZERO, ONE
! 158: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
! 159: $ ONE = ( 1.0d0, 0.0d0 ) )
! 160: DOUBLE PRECISION RZERO, RONE
! 161: PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0 )
! 162: * ..
! 163: * .. Local Scalars ..
! 164: COMPLEX*16 BETA, CDUM, S, TAU
! 165: DOUBLE PRECISION FOO, SAFMAX, SAFMIN, SMLNUM, ULP
! 166: INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
! 167: $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWKOPT
! 168: * ..
! 169: * .. External Functions ..
! 170: DOUBLE PRECISION DLAMCH
! 171: EXTERNAL DLAMCH
! 172: * ..
! 173: * .. External Subroutines ..
! 174: EXTERNAL DLABAD, ZCOPY, ZGEHRD, ZGEMM, ZLACPY, ZLAHQR,
! 175: $ ZLARF, ZLARFG, ZLASET, ZTREXC, ZUNMHR
! 176: * ..
! 177: * .. Intrinsic Functions ..
! 178: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, MAX, MIN
! 179: * ..
! 180: * .. Statement Functions ..
! 181: DOUBLE PRECISION CABS1
! 182: * ..
! 183: * .. Statement Function definitions ..
! 184: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
! 185: * ..
! 186: * .. Executable Statements ..
! 187: *
! 188: * ==== Estimate optimal workspace. ====
! 189: *
! 190: JW = MIN( NW, KBOT-KTOP+1 )
! 191: IF( JW.LE.2 ) THEN
! 192: LWKOPT = 1
! 193: ELSE
! 194: *
! 195: * ==== Workspace query call to ZGEHRD ====
! 196: *
! 197: CALL ZGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
! 198: LWK1 = INT( WORK( 1 ) )
! 199: *
! 200: * ==== Workspace query call to ZUNMHR ====
! 201: *
! 202: CALL ZUNMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
! 203: $ WORK, -1, INFO )
! 204: LWK2 = INT( WORK( 1 ) )
! 205: *
! 206: * ==== Optimal workspace ====
! 207: *
! 208: LWKOPT = JW + MAX( LWK1, LWK2 )
! 209: END IF
! 210: *
! 211: * ==== Quick return in case of workspace query. ====
! 212: *
! 213: IF( LWORK.EQ.-1 ) THEN
! 214: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
! 215: RETURN
! 216: END IF
! 217: *
! 218: * ==== Nothing to do ...
! 219: * ... for an empty active block ... ====
! 220: NS = 0
! 221: ND = 0
! 222: WORK( 1 ) = ONE
! 223: IF( KTOP.GT.KBOT )
! 224: $ RETURN
! 225: * ... nor for an empty deflation window. ====
! 226: IF( NW.LT.1 )
! 227: $ RETURN
! 228: *
! 229: * ==== Machine constants ====
! 230: *
! 231: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
! 232: SAFMAX = RONE / SAFMIN
! 233: CALL DLABAD( SAFMIN, SAFMAX )
! 234: ULP = DLAMCH( 'PRECISION' )
! 235: SMLNUM = SAFMIN*( DBLE( N ) / ULP )
! 236: *
! 237: * ==== Setup deflation window ====
! 238: *
! 239: JW = MIN( NW, KBOT-KTOP+1 )
! 240: KWTOP = KBOT - JW + 1
! 241: IF( KWTOP.EQ.KTOP ) THEN
! 242: S = ZERO
! 243: ELSE
! 244: S = H( KWTOP, KWTOP-1 )
! 245: END IF
! 246: *
! 247: IF( KBOT.EQ.KWTOP ) THEN
! 248: *
! 249: * ==== 1-by-1 deflation window: not much to do ====
! 250: *
! 251: SH( KWTOP ) = H( KWTOP, KWTOP )
! 252: NS = 1
! 253: ND = 0
! 254: IF( CABS1( S ).LE.MAX( SMLNUM, ULP*CABS1( H( KWTOP,
! 255: $ KWTOP ) ) ) ) THEN
! 256: NS = 0
! 257: ND = 1
! 258: IF( KWTOP.GT.KTOP )
! 259: $ H( KWTOP, KWTOP-1 ) = ZERO
! 260: END IF
! 261: WORK( 1 ) = ONE
! 262: RETURN
! 263: END IF
! 264: *
! 265: * ==== Convert to spike-triangular form. (In case of a
! 266: * . rare QR failure, this routine continues to do
! 267: * . aggressive early deflation using that part of
! 268: * . the deflation window that converged using INFQR
! 269: * . here and there to keep track.) ====
! 270: *
! 271: CALL ZLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
! 272: CALL ZCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
! 273: *
! 274: CALL ZLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
! 275: CALL ZLAHQR( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
! 276: $ JW, V, LDV, INFQR )
! 277: *
! 278: * ==== Deflation detection loop ====
! 279: *
! 280: NS = JW
! 281: ILST = INFQR + 1
! 282: DO 10 KNT = INFQR + 1, JW
! 283: *
! 284: * ==== Small spike tip deflation test ====
! 285: *
! 286: FOO = CABS1( T( NS, NS ) )
! 287: IF( FOO.EQ.RZERO )
! 288: $ FOO = CABS1( S )
! 289: IF( CABS1( S )*CABS1( V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) )
! 290: $ THEN
! 291: *
! 292: * ==== One more converged eigenvalue ====
! 293: *
! 294: NS = NS - 1
! 295: ELSE
! 296: *
! 297: * ==== One undeflatable eigenvalue. Move it up out of the
! 298: * . way. (ZTREXC can not fail in this case.) ====
! 299: *
! 300: IFST = NS
! 301: CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
! 302: ILST = ILST + 1
! 303: END IF
! 304: 10 CONTINUE
! 305: *
! 306: * ==== Return to Hessenberg form ====
! 307: *
! 308: IF( NS.EQ.0 )
! 309: $ S = ZERO
! 310: *
! 311: IF( NS.LT.JW ) THEN
! 312: *
! 313: * ==== sorting the diagonal of T improves accuracy for
! 314: * . graded matrices. ====
! 315: *
! 316: DO 30 I = INFQR + 1, NS
! 317: IFST = I
! 318: DO 20 J = I + 1, NS
! 319: IF( CABS1( T( J, J ) ).GT.CABS1( T( IFST, IFST ) ) )
! 320: $ IFST = J
! 321: 20 CONTINUE
! 322: ILST = I
! 323: IF( IFST.NE.ILST )
! 324: $ CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
! 325: 30 CONTINUE
! 326: END IF
! 327: *
! 328: * ==== Restore shift/eigenvalue array from T ====
! 329: *
! 330: DO 40 I = INFQR + 1, JW
! 331: SH( KWTOP+I-1 ) = T( I, I )
! 332: 40 CONTINUE
! 333: *
! 334: *
! 335: IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
! 336: IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
! 337: *
! 338: * ==== Reflect spike back into lower triangle ====
! 339: *
! 340: CALL ZCOPY( NS, V, LDV, WORK, 1 )
! 341: DO 50 I = 1, NS
! 342: WORK( I ) = DCONJG( WORK( I ) )
! 343: 50 CONTINUE
! 344: BETA = WORK( 1 )
! 345: CALL ZLARFG( NS, BETA, WORK( 2 ), 1, TAU )
! 346: WORK( 1 ) = ONE
! 347: *
! 348: CALL ZLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
! 349: *
! 350: CALL ZLARF( 'L', NS, JW, WORK, 1, DCONJG( TAU ), T, LDT,
! 351: $ WORK( JW+1 ) )
! 352: CALL ZLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
! 353: $ WORK( JW+1 ) )
! 354: CALL ZLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
! 355: $ WORK( JW+1 ) )
! 356: *
! 357: CALL ZGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
! 358: $ LWORK-JW, INFO )
! 359: END IF
! 360: *
! 361: * ==== Copy updated reduced window into place ====
! 362: *
! 363: IF( KWTOP.GT.1 )
! 364: $ H( KWTOP, KWTOP-1 ) = S*DCONJG( V( 1, 1 ) )
! 365: CALL ZLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
! 366: CALL ZCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
! 367: $ LDH+1 )
! 368: *
! 369: * ==== Accumulate orthogonal matrix in order update
! 370: * . H and Z, if requested. ====
! 371: *
! 372: IF( NS.GT.1 .AND. S.NE.ZERO )
! 373: $ CALL ZUNMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
! 374: $ WORK( JW+1 ), LWORK-JW, INFO )
! 375: *
! 376: * ==== Update vertical slab in H ====
! 377: *
! 378: IF( WANTT ) THEN
! 379: LTOP = 1
! 380: ELSE
! 381: LTOP = KTOP
! 382: END IF
! 383: DO 60 KROW = LTOP, KWTOP - 1, NV
! 384: KLN = MIN( NV, KWTOP-KROW )
! 385: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
! 386: $ LDH, V, LDV, ZERO, WV, LDWV )
! 387: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
! 388: 60 CONTINUE
! 389: *
! 390: * ==== Update horizontal slab in H ====
! 391: *
! 392: IF( WANTT ) THEN
! 393: DO 70 KCOL = KBOT + 1, N, NH
! 394: KLN = MIN( NH, N-KCOL+1 )
! 395: CALL ZGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
! 396: $ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
! 397: CALL ZLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
! 398: $ LDH )
! 399: 70 CONTINUE
! 400: END IF
! 401: *
! 402: * ==== Update vertical slab in Z ====
! 403: *
! 404: IF( WANTZ ) THEN
! 405: DO 80 KROW = ILOZ, IHIZ, NV
! 406: KLN = MIN( NV, IHIZ-KROW+1 )
! 407: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
! 408: $ LDZ, V, LDV, ZERO, WV, LDWV )
! 409: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
! 410: $ LDZ )
! 411: 80 CONTINUE
! 412: END IF
! 413: END IF
! 414: *
! 415: * ==== Return the number of deflations ... ====
! 416: *
! 417: ND = JW - NS
! 418: *
! 419: * ==== ... and the number of shifts. (Subtracting
! 420: * . INFQR from the spike length takes care
! 421: * . of the case of a rare QR failure while
! 422: * . calculating eigenvalues of the deflation
! 423: * . window.) ====
! 424: *
! 425: NS = NS - INFQR
! 426: *
! 427: * ==== Return optimal workspace. ====
! 428: *
! 429: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
! 430: *
! 431: * ==== End of ZLAQR2 ====
! 432: *
! 433: END
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