DGELSX(l)								 )								 DGELSX(l)

NAME

       DGELSX - routine is deprecated and has been replaced by routine DGELSY

SYNOPSIS

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

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

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  JPVT( * )

	   DOUBLE	  PRECISION A( LDA, * ), B( LDB, * ), WORK( * )

PURPOSE

       This  routine  is  deprecated  and  has	been replaced by routine DGELSY.  DGELSX 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
	       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) DOUBLE PRECISION 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

LAPACK version 3.0						   15 June 2000 							 DGELSX(l)