DLASDA(3)      LAPACK routine of NEC Numeric Library Collection      DLASDA(3)



NAME
       DLASDA

SYNOPSIS
       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)



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)
                     ICOMPQ is 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)
                     SMLSIZ is INTEGER
                    The maximum size of the subproblems at the bottom of the
                    computation tree.

           N         (input)
                     N is INTEGER
                    The row dimension of the upper bidiagonal matrix. This is
                    also the dimension of the main diagonal array D.

           SQRE      (input)
                     SQRE is 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)
                     D is 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)
                     E is DOUBLE PRECISION array, dimension ( M-1 )
                    Contains the subdiagonal entries of the bidiagonal matrix.
                    On exit, E has been destroyed.

           U         (output)
                     U is 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)
                     LDU is INTEGER, LDU = > N.
                    The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
                    GIVNUM, and Z.

           VT        (output)
                     VT is DOUBLE PRECISION array,
                    dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
                    if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
                    singular vector matrices of all subproblems at the bottom
                    level.

           K         (output)
                     K is 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)
                     DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ),
                    where NLVL = floor(log_2 (N/SMLSIZ))).

           DIFR      (output)
                     DIFR is 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)
                     Z is 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)
                     POLES is 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)
                     GIVPTR is 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)
                     GIVCOL is 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)
                     LDGCOL is INTEGER, LDGCOL = > N.
                    The leading dimension of arrays GIVCOL and PERM.

           PERM      (output)
                     PERM is 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)
                     GIVNUM is 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)
                     C is 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)
                     S is 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      (output)
                     WORK is DOUBLE PRECISION array, dimension
                    (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

           IWORK     (output)
                     IWORK is INTEGER array.
                    Dimension must be at least (7 * N).

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = 1, a singular value did not converge



LAPACK routine                  31 October 2017                      DLASDA(3)