

 Purpose
 =======

 LA_GELSX 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
 =========

  SUBROUTINE LA_GELSX( A, B, RANK=rank, JPVT=jpvt, RCOND=rcond, INFO=info )
       <type>(<wp>), INTENT( INOUT ) :: A(:,:), <rhs>
       INTEGER, INTENT(IN), OPTIONAL :: RANK
       INTEGER, INTENT(OUT), OPTIONAL :: JPVT(:)
       REAL(<wp>), INTENT(IN), OPTIONAL :: RCOND
       INTEGER, INTENT(OUT), OPTIONAL :: INFO
    where
       <type> ::= REAL | COMPLEX
       <wp>   ::= KIND(1.0) | KIND(1.0D0)
       <rhs>  ::= B(:,:) | B(:)

 ==========

 A    (input/output) Deither REAL or COMPLEX array, shape (:,:),
      SIZE(A,1) == m, SIZE(A,2) == n.
      On entry, the m-by-n matrix A.
      On exit, A has been overwritten by details of its
      complete orthogonal factorization.
      INFO = -1 if SIZE(A,1) < 0 or SIZE(A,2) < 0

 B    Optional (input/output) either REAL or COMPLEX array, shape either
      (:,:) or (:), size(B,1) or size(B) == size(A,1). SIZE(B,2) == 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.
      INFO = -2 if SIZE(B,1) /= max(SIZE(A,1), SIZE(A,2)) or SIZE(B,2) < 0
                    and if shape of B is (:,:) or
                if SIZE(B) /= max(SIZE(A,1), SIZE(A,2)) or SIZE(B,2) < 0
                   and if shape of B is (:)

 RANK Optional (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.

 JPVT Optional (input/output) INTEGER array, shape (:), SIZE(JPVT) == 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.
      INFO = -4 if SIZE(S) /= SIZE(A,2)

 RCOND Optional (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.

 INFO    (output) INTEGER
      = 0:  successful exit
      < 0:  if INFO = -i, the i-th argument had an illegal value
      If INFO is not present and an error occurs, then the program is
      terminated with an error message.

