DGELS(l)								 )								  DGELS(l)

NAME

       DGELS  -  solve overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ fac-
       torization of A

SYNOPSIS

       SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO )

	   CHARACTER	 TRANS

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

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

PURPOSE

       DGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factor-
       ization of A. It is assumed that A has full rank.  The following options are provided:

       1. If TRANS = 'N' and m >= n:  find the least squares solution of
	  an overdetermined system, i.e., solve the least squares problem
		       minimize || B - A*X ||.

       2. If TRANS = 'N' and m < n:  find the minimum norm solution of
	  an underdetermined system A * X = B.

       3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
	  an undetermined system A**T * X = B.

       4. If TRANS = 'T' and m < n:  find the least squares solution of
	  an overdetermined system, i.e., solve the least squares problem
		       minimize || B - A**T * X ||.

       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.

ARGUMENTS

       TRANS   (input) CHARACTER
	       = 'N': the linear system involves A;
	       = 'T': the linear system involves A**T.

       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 the 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, if M >= N, A is overwritten by details of its QR factorization as returned by DGEQRF; if M
	       <  N, A is overwritten by details of its LQ factorization as returned by DGELQF.

       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 matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'T'.
	       On exit, B is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
	       squares	solution  vectors;  the residual sum of squares for the solution in each column is given by the sum of squares of elements
	       N+1 to M in that column; if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors; if TRANS = 'T' and m
	       >=  n,  rows  1	to  M of B contain the minimum norm solution vectors; if TRANS = 'T' and m < n, rows 1 to M of B contain the least
	       squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of  squares  of  elements
	       M+1 to N in that column.

       LDB     (input) INTEGER
	       The leading dimension of the array B. LDB >= MAX(1,M,N).

       WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK.  LWORK >= max( 1, MN + max( MN, NRHS ) ).  For optimal performance, LWORK >= max( 1, MN + max( MN,
	       NRHS )*NB ).  where MN = min(M,N) and NB is the optimum block size.

	       If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of  the  WORK  array,  returns  this
	       value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

       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 							  DGELS(l)