SGELSS (3lapack)

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

SYNOPSIS

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

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

      REAL           RCOND

      REAL           A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

PURPOSE

  SGELSS 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) REAL 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).

  B       (input/output) REAL array, dimension (LDB,NRHS)
          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) REAL 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) REAL
          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 singular values which
          are greater than RCOND*S(1).

  WORK    (workspace/output) REAL array, dimension (LWORK)
          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*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) For good
          performance, 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.