sgelsy.f(3)							      LAPACK							       sgelsy.f(3)

NAME

       sgelsy.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sgelsy (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO)
	    SGELSY solves overdetermined or underdetermined systems for GE matrices

Function/Subroutine Documentation
   subroutine sgelsy (integerM, integerN, integerNRHS, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, integer,
       dimension( * )JPVT, realRCOND, integerRANK, real, dimension( * )WORK, integerLWORK, integerINFO)
	SGELSY solves overdetermined or underdetermined systems for GE matrices

       Purpose:

	    SGELSY 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**T [ inv(T11)*Q1**T*B ]
			    [	     0	       ]
	    where Q1 consists of the first RANK columns of Q.

	    This routine is basically identical to the original xGELSX except
	    three differences:
	      o The call to the subroutine xGEQPF has been substituted by the
		the call to the subroutine xGEQP3. This subroutine is a Blas-3
		version of the QR factorization with column pivoting.
	      o Matrix B (the right hand side) is updated with Blas-3.
	      o The permutation of matrix B (the right hand side) is faster and
		more simple.

       Parameters:
	   M

		     M is INTEGER
		     The number of rows of the matrix A.  M >= 0.

	   N

		     N is INTEGER
		     The number of columns of the matrix A.  N >= 0.

	   NRHS

		     NRHS is INTEGER
		     The number of right hand sides, i.e., the number of
		     columns of matrices B and X. NRHS >= 0.

	   A

		     A is 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

		     LDA is INTEGER
		     The leading dimension of the array A.  LDA >= max(1,M).

	   B

		     B is 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.

	   LDB

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

	   JPVT

		     JPVT is INTEGER array, dimension (N)
		     On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
		     to the front of AP, otherwise column i is a free column.
		     On exit, if JPVT(i) = k, then the i-th column of AP
		     was the k-th column of A.

	   RCOND

		     RCOND is 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

		     RANK is 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

		     WORK is REAL array, dimension (MAX(1,LWORK))
		     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

	   LWORK

		     LWORK is INTEGER
		     The dimension of the array WORK.
		     The unblocked strategy requires that:
			LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
		     where MN = min( M, N ).
		     The block algorithm requires that:
			LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
		     where NB is an upper bound on the blocksize returned
		     by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR,
		     and SORMRZ.

		     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

		     INFO is INTEGER
		     = 0: successful exit
		     < 0: If INFO = -i, the i-th argument had an illegal value.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Contributors:
	   A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
	    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
	    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

       Definition at line 204 of file sgelsy.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2							  Tue Sep 25 2012						       sgelsy.f(3)