CXML
        	    
LAPACK version 3.0
dgels(3)

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

SYNTAX
  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( * )

DESCRIPTION
  DGELS solves overdetermined or underdetermined real linear systems
  involving an M-by-N matrix A, or its transpose, using a QR or LQ
  factorization 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