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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 fac‐
       torization 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	 over‐
	       written	by details of its QR factorization as returned by DGE‐
	       QRF; if M <  N, A is overwritten by details of its  LQ  factor‐
	       ization 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 vec‐
	       tors, 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	 solu‐
	       tion  vectors; if TRANS = 'T' and m >= n, rows 1 to M of B con‐
	       tain 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 col‐
	       umn.

       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	 opti‐
	       mum 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)
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