SGELSX(l)	     LAPACK driver routine (version 1.1)	    SGELSX(l)

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 annihilated by orthog-
  onal 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