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RedHat 9 (Linux i386) - man page for sgelsy (redhat section l)

SGELSY(l)					)					SGELSY(l)

NAME
       SGELSY - compute the minimum-norm solution to a real linear least squares problem

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

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

	   REAL 	  RCOND

	   INTEGER	  JPVT( * )

	   REAL 	  A( LDA, * ), B( LDB, * ), WORK( * )

PURPOSE
       SGELSY computes the minimum-norm solution to a real linear least squares problem:     min-
       imize || 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 transforma-
       tions 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.

       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.

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

       B       (input/output) 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     (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 permuted to the front of  AP,
	       otherwise  column i is a free column.  On exit, if JPVT(i) = k, then the i-th col-
	       umn of AP was the k-th column of A.

       RCOND   (input) 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 piv-
	       oting 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/output) REAL array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) 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    (output) INTEGER
	       = 0: successful exit
	       < 0: If INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS
       Based on contributions by
	 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

LAPACK version 3.0			   15 June 2000 				SGELSY(l)


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