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sgelsx

 NAME
      SGELSX - 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 annihi-
      lated 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 sub-
              matrix R11.  This is the same as the order of the
              submatrix T11 in the complete orthogonal factoriza-
              tion 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