Annotation of rpl/lapack/lapack/dlasda.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
! 2: $ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
! 3: $ PERM, GIVNUM, C, S, WORK, IWORK, INFO )
! 4: *
! 5: * -- LAPACK auxiliary routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
! 12: * ..
! 13: * .. Array Arguments ..
! 14: INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
! 15: $ K( * ), PERM( LDGCOL, * )
! 16: DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
! 17: $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
! 18: $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
! 19: $ Z( LDU, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * Using a divide and conquer approach, DLASDA computes the singular
! 26: * value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
! 27: * B with diagonal D and offdiagonal E, where M = N + SQRE. The
! 28: * algorithm computes the singular values in the SVD B = U * S * VT.
! 29: * The orthogonal matrices U and VT are optionally computed in
! 30: * compact form.
! 31: *
! 32: * A related subroutine, DLASD0, computes the singular values and
! 33: * the singular vectors in explicit form.
! 34: *
! 35: * Arguments
! 36: * =========
! 37: *
! 38: * ICOMPQ (input) INTEGER
! 39: * Specifies whether singular vectors are to be computed
! 40: * in compact form, as follows
! 41: * = 0: Compute singular values only.
! 42: * = 1: Compute singular vectors of upper bidiagonal
! 43: * matrix in compact form.
! 44: *
! 45: * SMLSIZ (input) INTEGER
! 46: * The maximum size of the subproblems at the bottom of the
! 47: * computation tree.
! 48: *
! 49: * N (input) INTEGER
! 50: * The row dimension of the upper bidiagonal matrix. This is
! 51: * also the dimension of the main diagonal array D.
! 52: *
! 53: * SQRE (input) INTEGER
! 54: * Specifies the column dimension of the bidiagonal matrix.
! 55: * = 0: The bidiagonal matrix has column dimension M = N;
! 56: * = 1: The bidiagonal matrix has column dimension M = N + 1.
! 57: *
! 58: * D (input/output) DOUBLE PRECISION array, dimension ( N )
! 59: * On entry D contains the main diagonal of the bidiagonal
! 60: * matrix. On exit D, if INFO = 0, contains its singular values.
! 61: *
! 62: * E (input) DOUBLE PRECISION array, dimension ( M-1 )
! 63: * Contains the subdiagonal entries of the bidiagonal matrix.
! 64: * On exit, E has been destroyed.
! 65: *
! 66: * U (output) DOUBLE PRECISION array,
! 67: * dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
! 68: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
! 69: * singular vector matrices of all subproblems at the bottom
! 70: * level.
! 71: *
! 72: * LDU (input) INTEGER, LDU = > N.
! 73: * The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
! 74: * GIVNUM, and Z.
! 75: *
! 76: * VT (output) DOUBLE PRECISION array,
! 77: * dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
! 78: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right
! 79: * singular vector matrices of all subproblems at the bottom
! 80: * level.
! 81: *
! 82: * K (output) INTEGER array,
! 83: * dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
! 84: * If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
! 85: * secular equation on the computation tree.
! 86: *
! 87: * DIFL (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
! 88: * where NLVL = floor(log_2 (N/SMLSIZ))).
! 89: *
! 90: * DIFR (output) DOUBLE PRECISION array,
! 91: * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
! 92: * dimension ( N ) if ICOMPQ = 0.
! 93: * If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
! 94: * record distances between singular values on the I-th
! 95: * level and singular values on the (I -1)-th level, and
! 96: * DIFR(1:N, 2 * I ) contains the normalizing factors for
! 97: * the right singular vector matrix. See DLASD8 for details.
! 98: *
! 99: * Z (output) DOUBLE PRECISION array,
! 100: * dimension ( LDU, NLVL ) if ICOMPQ = 1 and
! 101: * dimension ( N ) if ICOMPQ = 0.
! 102: * The first K elements of Z(1, I) contain the components of
! 103: * the deflation-adjusted updating row vector for subproblems
! 104: * on the I-th level.
! 105: *
! 106: * POLES (output) DOUBLE PRECISION array,
! 107: * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
! 108: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
! 109: * POLES(1, 2*I) contain the new and old singular values
! 110: * involved in the secular equations on the I-th level.
! 111: *
! 112: * GIVPTR (output) INTEGER array,
! 113: * dimension ( N ) if ICOMPQ = 1, and not referenced if
! 114: * ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
! 115: * the number of Givens rotations performed on the I-th
! 116: * problem on the computation tree.
! 117: *
! 118: * GIVCOL (output) INTEGER array,
! 119: * dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
! 120: * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
! 121: * GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
! 122: * of Givens rotations performed on the I-th level on the
! 123: * computation tree.
! 124: *
! 125: * LDGCOL (input) INTEGER, LDGCOL = > N.
! 126: * The leading dimension of arrays GIVCOL and PERM.
! 127: *
! 128: * PERM (output) INTEGER array,
! 129: * dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
! 130: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
! 131: * permutations done on the I-th level of the computation tree.
! 132: *
! 133: * GIVNUM (output) DOUBLE PRECISION array,
! 134: * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not
! 135: * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
! 136: * GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
! 137: * values of Givens rotations performed on the I-th level on
! 138: * the computation tree.
! 139: *
! 140: * C (output) DOUBLE PRECISION array,
! 141: * dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
! 142: * If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
! 143: * C( I ) contains the C-value of a Givens rotation related to
! 144: * the right null space of the I-th subproblem.
! 145: *
! 146: * S (output) DOUBLE PRECISION array, dimension ( N ) if
! 147: * ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
! 148: * and the I-th subproblem is not square, on exit, S( I )
! 149: * contains the S-value of a Givens rotation related to
! 150: * the right null space of the I-th subproblem.
! 151: *
! 152: * WORK (workspace) DOUBLE PRECISION array, dimension
! 153: * (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
! 154: *
! 155: * IWORK (workspace) INTEGER array.
! 156: * Dimension must be at least (7 * N).
! 157: *
! 158: * INFO (output) INTEGER
! 159: * = 0: successful exit.
! 160: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 161: * > 0: if INFO = 1, an singular value did not converge
! 162: *
! 163: * Further Details
! 164: * ===============
! 165: *
! 166: * Based on contributions by
! 167: * Ming Gu and Huan Ren, Computer Science Division, University of
! 168: * California at Berkeley, USA
! 169: *
! 170: * =====================================================================
! 171: *
! 172: * .. Parameters ..
! 173: DOUBLE PRECISION ZERO, ONE
! 174: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 175: * ..
! 176: * .. Local Scalars ..
! 177: INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
! 178: $ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
! 179: $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
! 180: $ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
! 181: DOUBLE PRECISION ALPHA, BETA
! 182: * ..
! 183: * .. External Subroutines ..
! 184: EXTERNAL DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA
! 185: * ..
! 186: * .. Executable Statements ..
! 187: *
! 188: * Test the input parameters.
! 189: *
! 190: INFO = 0
! 191: *
! 192: IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
! 193: INFO = -1
! 194: ELSE IF( SMLSIZ.LT.3 ) THEN
! 195: INFO = -2
! 196: ELSE IF( N.LT.0 ) THEN
! 197: INFO = -3
! 198: ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
! 199: INFO = -4
! 200: ELSE IF( LDU.LT.( N+SQRE ) ) THEN
! 201: INFO = -8
! 202: ELSE IF( LDGCOL.LT.N ) THEN
! 203: INFO = -17
! 204: END IF
! 205: IF( INFO.NE.0 ) THEN
! 206: CALL XERBLA( 'DLASDA', -INFO )
! 207: RETURN
! 208: END IF
! 209: *
! 210: M = N + SQRE
! 211: *
! 212: * If the input matrix is too small, call DLASDQ to find the SVD.
! 213: *
! 214: IF( N.LE.SMLSIZ ) THEN
! 215: IF( ICOMPQ.EQ.0 ) THEN
! 216: CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU,
! 217: $ U, LDU, WORK, INFO )
! 218: ELSE
! 219: CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU,
! 220: $ U, LDU, WORK, INFO )
! 221: END IF
! 222: RETURN
! 223: END IF
! 224: *
! 225: * Book-keeping and set up the computation tree.
! 226: *
! 227: INODE = 1
! 228: NDIML = INODE + N
! 229: NDIMR = NDIML + N
! 230: IDXQ = NDIMR + N
! 231: IWK = IDXQ + N
! 232: *
! 233: NCC = 0
! 234: NRU = 0
! 235: *
! 236: SMLSZP = SMLSIZ + 1
! 237: VF = 1
! 238: VL = VF + M
! 239: NWORK1 = VL + M
! 240: NWORK2 = NWORK1 + SMLSZP*SMLSZP
! 241: *
! 242: CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
! 243: $ IWORK( NDIMR ), SMLSIZ )
! 244: *
! 245: * for the nodes on bottom level of the tree, solve
! 246: * their subproblems by DLASDQ.
! 247: *
! 248: NDB1 = ( ND+1 ) / 2
! 249: DO 30 I = NDB1, ND
! 250: *
! 251: * IC : center row of each node
! 252: * NL : number of rows of left subproblem
! 253: * NR : number of rows of right subproblem
! 254: * NLF: starting row of the left subproblem
! 255: * NRF: starting row of the right subproblem
! 256: *
! 257: I1 = I - 1
! 258: IC = IWORK( INODE+I1 )
! 259: NL = IWORK( NDIML+I1 )
! 260: NLP1 = NL + 1
! 261: NR = IWORK( NDIMR+I1 )
! 262: NLF = IC - NL
! 263: NRF = IC + 1
! 264: IDXQI = IDXQ + NLF - 2
! 265: VFI = VF + NLF - 1
! 266: VLI = VL + NLF - 1
! 267: SQREI = 1
! 268: IF( ICOMPQ.EQ.0 ) THEN
! 269: CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
! 270: $ SMLSZP )
! 271: CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),
! 272: $ E( NLF ), WORK( NWORK1 ), SMLSZP,
! 273: $ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,
! 274: $ WORK( NWORK2 ), INFO )
! 275: ITEMP = NWORK1 + NL*SMLSZP
! 276: CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
! 277: CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
! 278: ELSE
! 279: CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
! 280: CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU )
! 281: CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
! 282: $ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,
! 283: $ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )
! 284: CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
! 285: CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )
! 286: END IF
! 287: IF( INFO.NE.0 ) THEN
! 288: RETURN
! 289: END IF
! 290: DO 10 J = 1, NL
! 291: IWORK( IDXQI+J ) = J
! 292: 10 CONTINUE
! 293: IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN
! 294: SQREI = 0
! 295: ELSE
! 296: SQREI = 1
! 297: END IF
! 298: IDXQI = IDXQI + NLP1
! 299: VFI = VFI + NLP1
! 300: VLI = VLI + NLP1
! 301: NRP1 = NR + SQREI
! 302: IF( ICOMPQ.EQ.0 ) THEN
! 303: CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
! 304: $ SMLSZP )
! 305: CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),
! 306: $ E( NRF ), WORK( NWORK1 ), SMLSZP,
! 307: $ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,
! 308: $ WORK( NWORK2 ), INFO )
! 309: ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP
! 310: CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
! 311: CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
! 312: ELSE
! 313: CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
! 314: CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU )
! 315: CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
! 316: $ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,
! 317: $ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )
! 318: CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
! 319: CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )
! 320: END IF
! 321: IF( INFO.NE.0 ) THEN
! 322: RETURN
! 323: END IF
! 324: DO 20 J = 1, NR
! 325: IWORK( IDXQI+J ) = J
! 326: 20 CONTINUE
! 327: 30 CONTINUE
! 328: *
! 329: * Now conquer each subproblem bottom-up.
! 330: *
! 331: J = 2**NLVL
! 332: DO 50 LVL = NLVL, 1, -1
! 333: LVL2 = LVL*2 - 1
! 334: *
! 335: * Find the first node LF and last node LL on
! 336: * the current level LVL.
! 337: *
! 338: IF( LVL.EQ.1 ) THEN
! 339: LF = 1
! 340: LL = 1
! 341: ELSE
! 342: LF = 2**( LVL-1 )
! 343: LL = 2*LF - 1
! 344: END IF
! 345: DO 40 I = LF, LL
! 346: IM1 = I - 1
! 347: IC = IWORK( INODE+IM1 )
! 348: NL = IWORK( NDIML+IM1 )
! 349: NR = IWORK( NDIMR+IM1 )
! 350: NLF = IC - NL
! 351: NRF = IC + 1
! 352: IF( I.EQ.LL ) THEN
! 353: SQREI = SQRE
! 354: ELSE
! 355: SQREI = 1
! 356: END IF
! 357: VFI = VF + NLF - 1
! 358: VLI = VL + NLF - 1
! 359: IDXQI = IDXQ + NLF - 1
! 360: ALPHA = D( IC )
! 361: BETA = E( IC )
! 362: IF( ICOMPQ.EQ.0 ) THEN
! 363: CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
! 364: $ WORK( VFI ), WORK( VLI ), ALPHA, BETA,
! 365: $ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,
! 366: $ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,
! 367: $ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),
! 368: $ IWORK( IWK ), INFO )
! 369: ELSE
! 370: J = J - 1
! 371: CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
! 372: $ WORK( VFI ), WORK( VLI ), ALPHA, BETA,
! 373: $ IWORK( IDXQI ), PERM( NLF, LVL ),
! 374: $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
! 375: $ GIVNUM( NLF, LVL2 ), LDU,
! 376: $ POLES( NLF, LVL2 ), DIFL( NLF, LVL ),
! 377: $ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),
! 378: $ C( J ), S( J ), WORK( NWORK1 ),
! 379: $ IWORK( IWK ), INFO )
! 380: END IF
! 381: IF( INFO.NE.0 ) THEN
! 382: RETURN
! 383: END IF
! 384: 40 CONTINUE
! 385: 50 CONTINUE
! 386: *
! 387: RETURN
! 388: *
! 389: * End of DLASDA
! 390: *
! 391: END
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