Annotation of rpl/lapack/lapack/dlasd6.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
! 2: $ IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
! 3: $ LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
! 4: $ IWORK, INFO )
! 5: *
! 6: * -- LAPACK auxiliary routine (version 3.2) --
! 7: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 8: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 9: * November 2006
! 10: *
! 11: * .. Scalar Arguments ..
! 12: INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
! 13: $ NR, SQRE
! 14: DOUBLE PRECISION ALPHA, BETA, C, S
! 15: * ..
! 16: * .. Array Arguments ..
! 17: INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
! 18: $ PERM( * )
! 19: DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ),
! 20: $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
! 21: $ VF( * ), VL( * ), WORK( * ), Z( * )
! 22: * ..
! 23: *
! 24: * Purpose
! 25: * =======
! 26: *
! 27: * DLASD6 computes the SVD of an updated upper bidiagonal matrix B
! 28: * obtained by merging two smaller ones by appending a row. This
! 29: * routine is used only for the problem which requires all singular
! 30: * values and optionally singular vector matrices in factored form.
! 31: * B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
! 32: * A related subroutine, DLASD1, handles the case in which all singular
! 33: * values and singular vectors of the bidiagonal matrix are desired.
! 34: *
! 35: * DLASD6 computes the SVD as follows:
! 36: *
! 37: * ( D1(in) 0 0 0 )
! 38: * B = U(in) * ( Z1' a Z2' b ) * VT(in)
! 39: * ( 0 0 D2(in) 0 )
! 40: *
! 41: * = U(out) * ( D(out) 0) * VT(out)
! 42: *
! 43: * where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
! 44: * with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
! 45: * elsewhere; and the entry b is empty if SQRE = 0.
! 46: *
! 47: * The singular values of B can be computed using D1, D2, the first
! 48: * components of all the right singular vectors of the lower block, and
! 49: * the last components of all the right singular vectors of the upper
! 50: * block. These components are stored and updated in VF and VL,
! 51: * respectively, in DLASD6. Hence U and VT are not explicitly
! 52: * referenced.
! 53: *
! 54: * The singular values are stored in D. The algorithm consists of two
! 55: * stages:
! 56: *
! 57: * The first stage consists of deflating the size of the problem
! 58: * when there are multiple singular values or if there is a zero
! 59: * in the Z vector. For each such occurence the dimension of the
! 60: * secular equation problem is reduced by one. This stage is
! 61: * performed by the routine DLASD7.
! 62: *
! 63: * The second stage consists of calculating the updated
! 64: * singular values. This is done by finding the roots of the
! 65: * secular equation via the routine DLASD4 (as called by DLASD8).
! 66: * This routine also updates VF and VL and computes the distances
! 67: * between the updated singular values and the old singular
! 68: * values.
! 69: *
! 70: * DLASD6 is called from DLASDA.
! 71: *
! 72: * Arguments
! 73: * =========
! 74: *
! 75: * ICOMPQ (input) INTEGER
! 76: * Specifies whether singular vectors are to be computed in
! 77: * factored form:
! 78: * = 0: Compute singular values only.
! 79: * = 1: Compute singular vectors in factored form as well.
! 80: *
! 81: * NL (input) INTEGER
! 82: * The row dimension of the upper block. NL >= 1.
! 83: *
! 84: * NR (input) INTEGER
! 85: * The row dimension of the lower block. NR >= 1.
! 86: *
! 87: * SQRE (input) INTEGER
! 88: * = 0: the lower block is an NR-by-NR square matrix.
! 89: * = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
! 90: *
! 91: * The bidiagonal matrix has row dimension N = NL + NR + 1,
! 92: * and column dimension M = N + SQRE.
! 93: *
! 94: * D (input/output) DOUBLE PRECISION array, dimension ( NL+NR+1 ).
! 95: * On entry D(1:NL,1:NL) contains the singular values of the
! 96: * upper block, and D(NL+2:N) contains the singular values
! 97: * of the lower block. On exit D(1:N) contains the singular
! 98: * values of the modified matrix.
! 99: *
! 100: * VF (input/output) DOUBLE PRECISION array, dimension ( M )
! 101: * On entry, VF(1:NL+1) contains the first components of all
! 102: * right singular vectors of the upper block; and VF(NL+2:M)
! 103: * contains the first components of all right singular vectors
! 104: * of the lower block. On exit, VF contains the first components
! 105: * of all right singular vectors of the bidiagonal matrix.
! 106: *
! 107: * VL (input/output) DOUBLE PRECISION array, dimension ( M )
! 108: * On entry, VL(1:NL+1) contains the last components of all
! 109: * right singular vectors of the upper block; and VL(NL+2:M)
! 110: * contains the last components of all right singular vectors of
! 111: * the lower block. On exit, VL contains the last components of
! 112: * all right singular vectors of the bidiagonal matrix.
! 113: *
! 114: * ALPHA (input/output) DOUBLE PRECISION
! 115: * Contains the diagonal element associated with the added row.
! 116: *
! 117: * BETA (input/output) DOUBLE PRECISION
! 118: * Contains the off-diagonal element associated with the added
! 119: * row.
! 120: *
! 121: * IDXQ (output) INTEGER array, dimension ( N )
! 122: * This contains the permutation which will reintegrate the
! 123: * subproblem just solved back into sorted order, i.e.
! 124: * D( IDXQ( I = 1, N ) ) will be in ascending order.
! 125: *
! 126: * PERM (output) INTEGER array, dimension ( N )
! 127: * The permutations (from deflation and sorting) to be applied
! 128: * to each block. Not referenced if ICOMPQ = 0.
! 129: *
! 130: * GIVPTR (output) INTEGER
! 131: * The number of Givens rotations which took place in this
! 132: * subproblem. Not referenced if ICOMPQ = 0.
! 133: *
! 134: * GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
! 135: * Each pair of numbers indicates a pair of columns to take place
! 136: * in a Givens rotation. Not referenced if ICOMPQ = 0.
! 137: *
! 138: * LDGCOL (input) INTEGER
! 139: * leading dimension of GIVCOL, must be at least N.
! 140: *
! 141: * GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
! 142: * Each number indicates the C or S value to be used in the
! 143: * corresponding Givens rotation. Not referenced if ICOMPQ = 0.
! 144: *
! 145: * LDGNUM (input) INTEGER
! 146: * The leading dimension of GIVNUM and POLES, must be at least N.
! 147: *
! 148: * POLES (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
! 149: * On exit, POLES(1,*) is an array containing the new singular
! 150: * values obtained from solving the secular equation, and
! 151: * POLES(2,*) is an array containing the poles in the secular
! 152: * equation. Not referenced if ICOMPQ = 0.
! 153: *
! 154: * DIFL (output) DOUBLE PRECISION array, dimension ( N )
! 155: * On exit, DIFL(I) is the distance between I-th updated
! 156: * (undeflated) singular value and the I-th (undeflated) old
! 157: * singular value.
! 158: *
! 159: * DIFR (output) DOUBLE PRECISION array,
! 160: * dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
! 161: * dimension ( N ) if ICOMPQ = 0.
! 162: * On exit, DIFR(I, 1) is the distance between I-th updated
! 163: * (undeflated) singular value and the I+1-th (undeflated) old
! 164: * singular value.
! 165: *
! 166: * If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
! 167: * normalizing factors for the right singular vector matrix.
! 168: *
! 169: * See DLASD8 for details on DIFL and DIFR.
! 170: *
! 171: * Z (output) DOUBLE PRECISION array, dimension ( M )
! 172: * The first elements of this array contain the components
! 173: * of the deflation-adjusted updating row vector.
! 174: *
! 175: * K (output) INTEGER
! 176: * Contains the dimension of the non-deflated matrix,
! 177: * This is the order of the related secular equation. 1 <= K <=N.
! 178: *
! 179: * C (output) DOUBLE PRECISION
! 180: * C contains garbage if SQRE =0 and the C-value of a Givens
! 181: * rotation related to the right null space if SQRE = 1.
! 182: *
! 183: * S (output) DOUBLE PRECISION
! 184: * S contains garbage if SQRE =0 and the S-value of a Givens
! 185: * rotation related to the right null space if SQRE = 1.
! 186: *
! 187: * WORK (workspace) DOUBLE PRECISION array, dimension ( 4 * M )
! 188: *
! 189: * IWORK (workspace) INTEGER array, dimension ( 3 * N )
! 190: *
! 191: * INFO (output) INTEGER
! 192: * = 0: successful exit.
! 193: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 194: * > 0: if INFO = 1, an singular value did not converge
! 195: *
! 196: * Further Details
! 197: * ===============
! 198: *
! 199: * Based on contributions by
! 200: * Ming Gu and Huan Ren, Computer Science Division, University of
! 201: * California at Berkeley, USA
! 202: *
! 203: * =====================================================================
! 204: *
! 205: * .. Parameters ..
! 206: DOUBLE PRECISION ONE, ZERO
! 207: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 208: * ..
! 209: * .. Local Scalars ..
! 210: INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
! 211: $ N, N1, N2
! 212: DOUBLE PRECISION ORGNRM
! 213: * ..
! 214: * .. External Subroutines ..
! 215: EXTERNAL DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8, XERBLA
! 216: * ..
! 217: * .. Intrinsic Functions ..
! 218: INTRINSIC ABS, MAX
! 219: * ..
! 220: * .. Executable Statements ..
! 221: *
! 222: * Test the input parameters.
! 223: *
! 224: INFO = 0
! 225: N = NL + NR + 1
! 226: M = N + SQRE
! 227: *
! 228: IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
! 229: INFO = -1
! 230: ELSE IF( NL.LT.1 ) THEN
! 231: INFO = -2
! 232: ELSE IF( NR.LT.1 ) THEN
! 233: INFO = -3
! 234: ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
! 235: INFO = -4
! 236: ELSE IF( LDGCOL.LT.N ) THEN
! 237: INFO = -14
! 238: ELSE IF( LDGNUM.LT.N ) THEN
! 239: INFO = -16
! 240: END IF
! 241: IF( INFO.NE.0 ) THEN
! 242: CALL XERBLA( 'DLASD6', -INFO )
! 243: RETURN
! 244: END IF
! 245: *
! 246: * The following values are for bookkeeping purposes only. They are
! 247: * integer pointers which indicate the portion of the workspace
! 248: * used by a particular array in DLASD7 and DLASD8.
! 249: *
! 250: ISIGMA = 1
! 251: IW = ISIGMA + N
! 252: IVFW = IW + M
! 253: IVLW = IVFW + M
! 254: *
! 255: IDX = 1
! 256: IDXC = IDX + N
! 257: IDXP = IDXC + N
! 258: *
! 259: * Scale.
! 260: *
! 261: ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
! 262: D( NL+1 ) = ZERO
! 263: DO 10 I = 1, N
! 264: IF( ABS( D( I ) ).GT.ORGNRM ) THEN
! 265: ORGNRM = ABS( D( I ) )
! 266: END IF
! 267: 10 CONTINUE
! 268: CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
! 269: ALPHA = ALPHA / ORGNRM
! 270: BETA = BETA / ORGNRM
! 271: *
! 272: * Sort and Deflate singular values.
! 273: *
! 274: CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
! 275: $ WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
! 276: $ WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
! 277: $ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
! 278: $ INFO )
! 279: *
! 280: * Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
! 281: *
! 282: CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
! 283: $ WORK( ISIGMA ), WORK( IW ), INFO )
! 284: *
! 285: * Save the poles if ICOMPQ = 1.
! 286: *
! 287: IF( ICOMPQ.EQ.1 ) THEN
! 288: CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 )
! 289: CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
! 290: END IF
! 291: *
! 292: * Unscale.
! 293: *
! 294: CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
! 295: *
! 296: * Prepare the IDXQ sorting permutation.
! 297: *
! 298: N1 = K
! 299: N2 = N - K
! 300: CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
! 301: *
! 302: RETURN
! 303: *
! 304: * End of DLASD6
! 305: *
! 306: END
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