NAME

       lasq1 - lasq1: dqds step

SYNOPSIS

   Functions
       subroutine dlasq1 (n, d, e, work, info)
           DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by
           sbdsqr.
       subroutine slasq1 (n, d, e, work, info)
           SLASQ1 computes the singular values of a real square bidiagonal matrix. Used by
           sbdsqr.

Detailed Description

Function Documentation

   subroutine dlasq1 (integer n, double precision, dimension( * ) d, double precision, dimension(
       * ) e, double precision, dimension( * ) work, integer info)
       DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.

       Purpose:

            DLASQ1 computes the singular values of a real N-by-N bidiagonal
            matrix with diagonal D and off-diagonal E. The singular values
            are computed to high relative accuracy, in the absence of
            denormalization, underflow and overflow. The algorithm was first
            presented in

            'Accurate singular values and differential qd algorithms' by K. V.
            Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
            1994,

            and the present implementation is described in 'An implementation of
            the dqds Algorithm (Positive Case)', LAPACK Working Note.

       Parameters
           N

                     N is INTEGER
                   The number of rows and columns in the matrix. N >= 0.

           D

                     D is DOUBLE PRECISION array, dimension (N)
                   On entry, D contains the diagonal elements of the
                   bidiagonal matrix whose SVD is desired. On normal exit,
                   D contains the singular values in decreasing order.

           E

                     E is DOUBLE PRECISION array, dimension (N)
                   On entry, elements E(1:N-1) contain the off-diagonal elements
                   of the bidiagonal matrix whose SVD is desired.
                   On exit, E is overwritten.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (4*N)

           INFO

                     INFO is INTEGER
                   = 0: successful exit
                   < 0: if INFO = -i, the i-th argument had an illegal value
                   > 0: the algorithm failed
                        = 1, a split was marked by a positive value in E
                        = 2, current block of Z not diagonalized after 100*N
                             iterations (in inner while loop)  On exit D and E
                             represent a matrix with the same singular values
                             which the calling subroutine could use to finish the
                             computation, or even feed back into DLASQ1
                        = 3, termination criterion of outer while loop not met
                             (program created more than N unreduced blocks)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine slasq1 (integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension(
       * ) work, integer info)
       SLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.

       Purpose:

            SLASQ1 computes the singular values of a real N-by-N bidiagonal
            matrix with diagonal D and off-diagonal E. The singular values
            are computed to high relative accuracy, in the absence of
            denormalization, underflow and overflow. The algorithm was first
            presented in

            'Accurate singular values and differential qd algorithms' by K. V.
            Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
            1994,

            and the present implementation is described in 'An implementation of
            the dqds Algorithm (Positive Case)', LAPACK Working Note.

       Parameters
           N

                     N is INTEGER
                   The number of rows and columns in the matrix. N >= 0.

           D

                     D is REAL array, dimension (N)
                   On entry, D contains the diagonal elements of the
                   bidiagonal matrix whose SVD is desired. On normal exit,
                   D contains the singular values in decreasing order.

           E

                     E is REAL array, dimension (N)
                   On entry, elements E(1:N-1) contain the off-diagonal elements
                   of the bidiagonal matrix whose SVD is desired.
                   On exit, E is overwritten.

           WORK

                     WORK is REAL array, dimension (4*N)

           INFO

                     INFO is INTEGER
                   = 0: successful exit
                   < 0: if INFO = -i, the i-th argument had an illegal value
                   > 0: the algorithm failed
                        = 1, a split was marked by a positive value in E
                        = 2, current block of Z not diagonalized after 100*N
                             iterations (in inner while loop)  On exit D and E
                             represent a matrix with the same singular values
                             which the calling subroutine could use to finish the
                             computation, or even feed back into SLASQ1
                        = 3, termination criterion of outer while loop not met
                             (program created more than N unreduced blocks)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

Author

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