SGEQLF  -  compute  a  QL  factorization  of a real M-by-N
       matrix A


SYNOPSIS

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

           INTEGER        INFO, LDA, LWORK, M, N

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


PURPOSE

       SGEQLF computes a QL factorization of a real M-by-N matrix
       A: A = Q * L.


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, if m >=
               n,  the  lower  triangle  of  the  subarray   A(m-
               n+1:m,1:n)  contains  the  N-by-N lower triangular
               matrix L; if m <= n, the elements on and below the
               (n-m)-th  superdiagonal  contain  the M-by-N lower
               trapezoidal matrix L; the remaining elements, 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,N).   For optimum performance LWORK >= N*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(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is  stored
       on exit in A(1:m-k+i-1,n-k+i), and tau in TAU(i).


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