DLASD1(3)      LAPACK routine of NEC Numeric Library Collection      DLASD1(3)



NAME
       DLASD1

SYNOPSIS
       SUBROUTINE DLASD1 (NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
           IDXQ, IWORK, WORK, INFO)



PURPOSE
            DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
            where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.

            A related subroutine DLASD7 handles the case in which the singular
            values (and the singular vectors in factored form) are desired.

            DLASD1 computes the SVD as follows:

                          ( D1(in)    0    0       0 )
              B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
                          (   0       0   D2(in)   0 )

                = U(out) * ( D(out) 0) * VT(out)

            where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
            with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
            elsewhere; and the entry b is empty if SQRE = 0.

            The left singular vectors of the original matrix are stored in U, and
            the transpose of the right singular vectors are stored in VT, and the
            singular values are in D.  The algorithm consists of three stages:

               The first stage consists of deflating the size of the problem
               when there are multiple singular values or when there are zeros in
               the Z vector.  For each such occurence the dimension of the
               secular equation problem is reduced by one.  This stage is
               performed by the routine DLASD2.

               The second stage consists of calculating the updated
               singular values. This is done by finding the square roots of the
               roots of the secular equation via the routine DLASD4 (as called
               by DLASD3). This routine also calculates the singular vectors of
               the current problem.

               The final stage consists of computing the updated singular vectors
               directly using the updated singular values.  The singular vectors
               for the current problem are multiplied with the singular vectors
               from the overall problem.




ARGUMENTS
           NL        (input)
                     NL is INTEGER
                    The row dimension of the upper block.  NL >= 1.

           NR        (input)
                     NR is INTEGER
                    The row dimension of the lower block.  NR >= 1.

           SQRE      (input)
                     SQRE is INTEGER
                    = 0: the lower block is an NR-by-NR square matrix.
                    = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

                    The bidiagonal matrix has row dimension N = NL + NR + 1,
                    and column dimension M = N + SQRE.

           D         (input/output)
                     D is DOUBLE PRECISION array,
                                   dimension (N = NL+NR+1).
                    On entry D(1:NL,1:NL) contains the singular values of the
                    upper block; and D(NL+2:N) contains the singular values of
                    the lower block. On exit D(1:N) contains the singular values
                    of the modified matrix.

           ALPHA     (input/output)
                     ALPHA is DOUBLE PRECISION
                    Contains the diagonal element associated with the added row.

           BETA      (input/output)
                     BETA is DOUBLE PRECISION
                    Contains the off-diagonal element associated with the added
                    row.

           U         (input/output)
                     U is DOUBLE PRECISION array, dimension(LDU,N)
                    On entry U(1:NL, 1:NL) contains the left singular vectors of
                    the upper block; U(NL+2:N, NL+2:N) contains the left singular
                    vectors of the lower block. On exit U contains the left
                    singular vectors of the bidiagonal matrix.

           LDU       (input)
                     LDU is INTEGER
                    The leading dimension of the array U.  LDU >= max( 1, N ).

           VT        (input/output)
                     VT is DOUBLE PRECISION array, dimension(LDVT,M)
                    where M = N + SQRE.
                    On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
                    vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
                    the right singular vectors of the lower block. On exit
                    VT**T contains the right singular vectors of the
                    bidiagonal matrix.

           LDVT      (input)
                     LDVT is INTEGER
                    The leading dimension of the array VT.  LDVT >= max( 1, M ).

           IDXQ      (output)
                     IDXQ is INTEGER array, dimension(N)
                    This contains the permutation which will reintegrate the
                    subproblem just solved back into sorted order, i.e.
                    D( IDXQ( I = 1, N ) ) will be in ascending order.

           IWORK     (output)
                     IWORK is INTEGER array, dimension( 4 * N )

           WORK      (output)
                     WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )

           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                      DLASD1(3)