dlasda.f

          SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
         $                   DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
         $                   PERM, GIVNUM, C, S, WORK, IWORK, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
    *     ..
    *     .. Array Arguments ..
          INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
         $                   K( * ), PERM( LDGCOL, * )
          DOUBLE PRECISION   C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
         $                   E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
         $                   S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
         $                   Z( LDU, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  Using a divide and conquer approach, DLASDA computes the singular
    *  value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
    *  B with diagonal D and offdiagonal E, where M = N + SQRE. The
    *  algorithm computes the singular values in the SVD B = U * S * VT.
    *  The orthogonal matrices U and VT are optionally computed in
    *  compact form.
    *
    *  A related subroutine, DLASD0, computes the singular values and
    *  the singular vectors in explicit form.
    *
    *  Arguments
    *  =========
    *
    *  ICOMPQ (input) INTEGER
    *         Specifies whether singular vectors are to be computed
    *         in compact form, as follows
    *         = 0: Compute singular values only.
    *         = 1: Compute singular vectors of upper bidiagonal
    *              matrix in compact form.
    *
    *  SMLSIZ (input) INTEGER
    *         The maximum size of the subproblems at the bottom of the
    *         computation tree.
    *
    *  N      (input) INTEGER
    *         The row dimension of the upper bidiagonal matrix. This is
    *         also the dimension of the main diagonal array D.
    *
    *  SQRE   (input) INTEGER
    *         Specifies the column dimension of the bidiagonal matrix.
    *         = 0: The bidiagonal matrix has column dimension M = N;
    *         = 1: The bidiagonal matrix has column dimension M = N + 1.
    *
    *  D      (input/output) DOUBLE PRECISION array, dimension ( N )
    *         On entry D contains the main diagonal of the bidiagonal
    *         matrix. On exit D, if INFO = 0, contains its singular values.
    *
    *  E      (input) DOUBLE PRECISION array, dimension ( M-1 )
    *         Contains the subdiagonal entries of the bidiagonal matrix.
    *         On exit, E has been destroyed.
    *
    *  U      (output) DOUBLE PRECISION array,
    *         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
    *         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
    *         singular vector matrices of all subproblems at the bottom
    *         level.
    *
    *  LDU    (input) INTEGER, LDU = > N.
    *         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
    *         GIVNUM, and Z.
    *
    *  VT     (output) DOUBLE PRECISION array,
    *         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
    *         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right
    *         singular vector matrices of all subproblems at the bottom
    *         level.
    *
    *  K      (output) INTEGER array,
    *         dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
    *         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
    *         secular equation on the computation tree.
    *
    *  DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
    *         where NLVL = floor(log_2 (N/SMLSIZ))).
    *
    *  DIFR   (output) DOUBLE PRECISION array,
    *                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
    *                  dimension ( N ) if ICOMPQ = 0.
    *         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
    *         record distances between singular values on the I-th
    *         level and singular values on the (I -1)-th level, and
    *         DIFR(1:N, 2 * I ) contains the normalizing factors for
    *         the right singular vector matrix. See DLASD8 for details.
    *
    *  Z      (output) DOUBLE PRECISION array,
    *                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
    *                  dimension ( N ) if ICOMPQ = 0.
    *         The first K elements of Z(1, I) contain the components of
    *         the deflation-adjusted updating row vector for subproblems
    *         on the I-th level.
    *
    *  POLES  (output) DOUBLE PRECISION array,
    *         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
    *         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
    *         POLES(1, 2*I) contain  the new and old singular values
    *         involved in the secular equations on the I-th level.
    *
    *  GIVPTR (output) INTEGER array,
    *         dimension ( N ) if ICOMPQ = 1, and not referenced if
    *         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
    *         the number of Givens rotations performed on the I-th
    *         problem on the computation tree.
    *
    *  GIVCOL (output) INTEGER array,
    *         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
    *         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
    *         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
    *         of Givens rotations performed on the I-th level on the
    *         computation tree.
    *
    *  LDGCOL (input) INTEGER, LDGCOL = > N.
    *         The leading dimension of arrays GIVCOL and PERM.
    *
    *  PERM   (output) INTEGER array,
    *         dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
    *         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
    *         permutations done on the I-th level of the computation tree.
    *
    *  GIVNUM (output) DOUBLE PRECISION array,
    *         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
    *         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
    *         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
    *         values of Givens rotations performed on the I-th level on
    *         the computation tree.
    *
    *  C      (output) DOUBLE PRECISION array,
    *         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
    *         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
    *         C( I ) contains the C-value of a Givens rotation related to
    *         the right null space of the I-th subproblem.
    *
    *  S      (output) DOUBLE PRECISION array, dimension ( N ) if
    *         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
    *         and the I-th subproblem is not square, on exit, S( I )
    *         contains the S-value of a Givens rotation related to
    *         the right null space of the I-th subproblem.
    *
    *  WORK   (workspace) DOUBLE PRECISION array, dimension
    *         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
    *
    *  IWORK  (workspace) INTEGER array.
    *         Dimension must be at least (7 * N).
    *
    *  INFO   (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          > 0:  if INFO = 1, an singular value did not converge
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Ming Gu and Huan Ren, Computer Science Division, University of
    *     California at Berkeley, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
         $                   J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
         $                   NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
         $                   NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
          DOUBLE PRECISION   ALPHA, BETA
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
             INFO = -1
          ELSE IF( SMLSIZ.LT.3 ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
             INFO = -4
          ELSE IF( LDU.LT.( N+SQRE ) ) THEN
             INFO = -8
          ELSE IF( LDGCOL.LT.N ) THEN
             INFO = -17
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DLASDA', -INFO )
             RETURN
          END IF
    *
          M = N + SQRE
    *
    *     If the input matrix is too small, call DLASDQ to find the SVD.
    *
          IF( N.LE.SMLSIZ ) THEN
             IF( ICOMPQ.EQ.0 ) THEN
                CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU,
         $                   U, LDU, WORK, INFO )
             ELSE
                CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU,
         $                   U, LDU, WORK, INFO )
             END IF
             RETURN
          END IF
    *
    *     Book-keeping and  set up the computation tree.
    *
          INODE = 1
          NDIML = INODE + N
          NDIMR = NDIML + N
          IDXQ = NDIMR + N
          IWK = IDXQ + N
    *
          NCC = 0
          NRU = 0
    *
          SMLSZP = SMLSIZ + 1
          VF = 1
          VL = VF + M
          NWORK1 = VL + M
          NWORK2 = NWORK1 + SMLSZP*SMLSZP
    *
          CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
         $             IWORK( NDIMR ), SMLSIZ )
    *
    *     for the nodes on bottom level of the tree, solve
    *     their subproblems by DLASDQ.
    *
          NDB1 = ( ND+1 ) / 2
          DO 30 I = NDB1, ND
    *
    *        IC : center row of each node
    *        NL : number of rows of left  subproblem
    *        NR : number of rows of right subproblem
    *        NLF: starting row of the left   subproblem
    *        NRF: starting row of the right  subproblem
    *
             I1 = I - 1
             IC = IWORK( INODE+I1 )
             NL = IWORK( NDIML+I1 )
             NLP1 = NL + 1
             NR = IWORK( NDIMR+I1 )
             NLF = IC - NL
             NRF = IC + 1
             IDXQI = IDXQ + NLF - 2
             VFI = VF + NLF - 1
             VLI = VL + NLF - 1
             SQREI = 1
             IF( ICOMPQ.EQ.0 ) THEN
                CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
         $                   SMLSZP )
                CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),
         $                   E( NLF ), WORK( NWORK1 ), SMLSZP,
         $                   WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,
         $                   WORK( NWORK2 ), INFO )
                ITEMP = NWORK1 + NL*SMLSZP
                CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
                CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
             ELSE
                CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
                CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU )
                CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
         $                   E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,
         $                   U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )
                CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
                CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )
             END IF
             IF( INFO.NE.0 ) THEN
                RETURN
             END IF
             DO 10 J = 1, NL
                IWORK( IDXQI+J ) = J
       10    CONTINUE
             IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN
                SQREI = 0
             ELSE
                SQREI = 1
             END IF
             IDXQI = IDXQI + NLP1
             VFI = VFI + NLP1
             VLI = VLI + NLP1
             NRP1 = NR + SQREI
             IF( ICOMPQ.EQ.0 ) THEN
                CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
         $                   SMLSZP )
                CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),
         $                   E( NRF ), WORK( NWORK1 ), SMLSZP,
         $                   WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,
         $                   WORK( NWORK2 ), INFO )
                ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP
                CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
                CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
             ELSE
                CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
                CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU )
                CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
         $                   E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,
         $                   U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )
                CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
                CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )
             END IF
             IF( INFO.NE.0 ) THEN
                RETURN
             END IF
             DO 20 J = 1, NR
                IWORK( IDXQI+J ) = J
       20    CONTINUE
       30 CONTINUE
    *
    *     Now conquer each subproblem bottom-up.
    *
          J = 2**NLVL
          DO 50 LVL = NLVL, 1, -1
             LVL2 = LVL*2 - 1
    *
    *        Find the first node LF and last node LL on
    *        the current level LVL.
    *
             IF( LVL.EQ.1 ) THEN
                LF = 1
                LL = 1
             ELSE
                LF = 2**( LVL-1 )
                LL = 2*LF - 1
             END IF
             DO 40 I = LF, LL
                IM1 = I - 1
                IC = IWORK( INODE+IM1 )
                NL = IWORK( NDIML+IM1 )
                NR = IWORK( NDIMR+IM1 )
                NLF = IC - NL
                NRF = IC + 1
                IF( I.EQ.LL ) THEN
                   SQREI = SQRE
                ELSE
                   SQREI = 1
                END IF
                VFI = VF + NLF - 1
                VLI = VL + NLF - 1
                IDXQI = IDXQ + NLF - 1
                ALPHA = D( IC )
                BETA = E( IC )
                IF( ICOMPQ.EQ.0 ) THEN
                   CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
         $                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,
         $                      IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,
         $                      LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,
         $                      K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),
         $                      IWORK( IWK ), INFO )
                ELSE
                   J = J - 1
                   CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
         $                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,
         $                      IWORK( IDXQI ), PERM( NLF, LVL ),
         $                      GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
         $                      GIVNUM( NLF, LVL2 ), LDU,
         $                      POLES( NLF, LVL2 ), DIFL( NLF, LVL ),
         $                      DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),
         $                      C( J ), S( J ), WORK( NWORK1 ),
         $                      IWORK( IWK ), INFO )
                END IF
                IF( INFO.NE.0 ) THEN
                   RETURN
                END IF
       40    CONTINUE
       50 CONTINUE
    *
          RETURN
    *
    *     End of DLASDA
    *
          END