SGELSX (3lapack)

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

SYNOPSIS

  SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
                     INFO )

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

      REAL           RCOND

      INTEGER        JPVT( * )

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

PURPOSE

  SGELSX computes the minimum-norm solution to a real linear least squares
  problem:
      minimize || A * X - B ||
  using a complete orthogonal factorization 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 routine first computes a QR factorization with column pivoting:
      A * P = Q * [ R11 R12 ]
                  [  0  R22 ]
  with R11 defined as the largest leading submatrix whose estimated condition
  number is less than 1/RCOND.  The order of R11, RANK, is the effective rank
  of A.

  Then, R22 is considered to be negligible, and R12 is annihilated by
  orthogonal transformations from the right, arriving at the complete
  orthogonal factorization:
     A * P = Q * [ T11 0 ] * Z
                 [  0  0 ]
  The minimum-norm solution is then
     X = P * Z' [ inv(T11)*Q1'*B ]
                [        0       ]
  where Q1 consists of the first RANK columns of Q.

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
          matrices B and X. NRHS >= 0.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.  On exit, A has been overwritten by
          details of its complete orthogonal factorization.

  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, 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,M,N).

  JPVT    (input/output) INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial
          column, otherwise it is a free column.  Before the QR factorization
          of A, all initial columns are permuted to the leading positions;
          only the remaining free columns are moved as a result of column
          pivoting during the factorization.  On exit, if JPVT(i) = k, then
          the i-th column of A*P was the k-th column of A.

  RCOND   (input) REAL
          RCOND is used to determine the effective rank of A, which is
          defined as the order of the largest leading triangular submatrix
          R11 in the QR factorization with pivoting of A, whose estimated
          condition number < 1/RCOND.

  RANK    (output) INTEGER
          The effective rank of A, i.e., the order of the submatrix R11.
          This is the same as the order of the submatrix T11 in the complete
          orthogonal factorization of A.

  WORK    (workspace) REAL array, dimension
          (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value