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dgelss

 NAME
      DGELSS - compute the minimum norm solution to a real linear
      least squares problem

 SYNOPSIS
      SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
                         RANK, WORK, LWORK, INFO )

          INTEGER        INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

          DOUBLE         PRECISION RCOND

          DOUBLE         PRECISION A( LDA, * ), B( LDB, * ), S( *
                         ), WORK( * )

 PURPOSE
      DGELSS computes the minimum norm solution to a real linear
      least squares problem:

      Minimize 2-norm(| b - A*x |).

      using the singular value decomposition (SVD) of A. A is an
      M-by-N matrix which may be rank-deficient.

      Several right hand side vectors b and solution vectors x can
      be handled in a single call; they are stored as the columns
      of the M-by-NRHS right hand side matrix B and the N-by-NRHS
      solution matrix X.

      The effective rank of A is determined by treating as zero
      those singular values which are less than RCOND times the
      largest singular value.

 ARGUMENTS
      M       (input) INTEGER
              The number of rows of the matrix A. M >= 0.

      N       (input) INTEGER
              The number of columns of the matrix A. N >= 0.

      NRHS    (input) INTEGER
              The number of right hand sides, i.e., the number of
              columns of the matrices B and X. NRHS >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.  On exit, the first
              min(m,n) rows of A are overwritten with its right
              singular vectors, stored rowwise.

      LDA     (input) INTEGER
              The leading dimension of the array A.  LDA >=

              max(1,M).

 (LDB,NRHS)
      B       (input/output) DOUBLE PRECISION array, dimension
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution
              matrix X.  If m >= n and RANK = n, the residual
              sum-of-squares for the solution in the i-th column
              is given by the sum of squares of elements n+1:m in
              that column.

      LDB     (input) INTEGER
              The leading dimension of the array B. LDB >=
              max(1,MAX(M,N)).

      S       (output) DOUBLE PRECISION array, dimension (min(M,N))
              The singular values of A in decreasing order.  The
              condition number of A in the 2-norm =
              S(1)/S(min(m,n)).

      RCOND   (input) DOUBLE PRECISION
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as
              zero.  If RCOND $<$ 0, machine precision is used
              instead.

      RANK    (output) INTEGER
              The effective rank of A, i.e., the number of singu-
              lar values which are greater than RCOND*S(1).

 (LWORK)
      WORK    (workspace/output) DOUBLE PRECISION array, dimension
              On exit, if INFO = 0, WORK(1) returns the optimal
              LWORK.

      LWORK   (input) INTEGER
              The dimension of the array WORK. LWORK >= 1, and
              also: LWORK >= 3*N+MAX(2*N,NRHS,M) if M >= N, LWORK
              >= 3*M+MAX(2*M,NRHS,N) if M < N.  For good perfor-
              mance, LWORK should generally be larger.

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
              value.
              > 0:  the algorithm for computing the SVD failed to
              converge; if INFO = i, i off-diagonal elements of an
              intermediate bidiagonal form did not converge to
              zero.