SGELSY(3)             LAPACK driver routine (version 3.1)            SGELSY(3)



NAME
       SGELSY - the minimum-norm solution to a real linear least squares prob-
       lem

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:
           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 con-
       dition 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' [ 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 column 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 subma-
               trix R11 in the QR factorization  with  pivoting  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  com-
               plete orthogonal factorization of A.

       WORK    (workspace/output) REAL array, dimension (MAX(1,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 driver routine (version 3.N1o)vember 2006                       SGELSY(3)