SGELQF  -  compute  an  LQ  factorization of a real M-by-N
       matrix A


SYNOPSIS

       SUBROUTINE SGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )

           INTEGER        INFO, LDA, LWORK, M, N

           REAL           A( LDA, * ), TAU( * ), WORK( * )


PURPOSE

       SGELQF computes an  LQ  factorization  of  a  real  M-by-N
       matrix A: A = L * Q.


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.

       A       (input/output) REAL array, dimension (LDA,N)
               On  entry, the M-by-N matrix A.  On exit, the ele­
               ments on and below the diagonal of the array  con­
               tain  the m-by-min(m,n) lower trapezoidal matrix L
               (L is lower triangular if m <=  n);  the  elements
               above  the diagonal, with the array TAU, represent
               the orthogonal matrix Q as a product of elementary
               reflectors (see Further Details).  LDA     (input)
               INTEGER The leading dimension of the array A.  LDA
               >= max(1,M).

       TAU     (output) REAL array, dimension (min(M,N))
               The  scalar  factors  of the elementary reflectors
               (see Further Details).

       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.   LWORK  >=
               max(1,M).  For optimum performance LWORK >=  M*NB,
               where NB is the optimal blocksize.

               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.

               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value


FURTHER DETAILS

       The matrix Q is represented as  a  product  of  elementary
       reflectors

          Q = H(k) . . . H(2) H(1), where k = min(m,n).

       Each H(i) has the form

          H(i) = I - tau * v * v'

       where tau is a real scalar, and v is a real vector with
       v(1:i-1)  =  0 and v(i) = 1; v(i+1:n) is stored on exit in
       A(i,i+1:n), and tau in TAU(i).


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