SGEQRF  -  compute  a  QR  factorization  of a real M-by-N
       matrix A


SYNOPSIS

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

           INTEGER        INFO, LDA, LWORK, M, N

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


PURPOSE

       SGEQRF computes a QR factorization of a real M-by-N matrix
       A: A = Q * R.


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 above the diagonal of the array  con­
               tain  the min(M,N)-by-N upper trapezoidal matrix R
               (R is upper triangular if m >=  n);  the  elements
               below  the diagonal, with the array TAU, represent
               the orthogonal matrix Q as a product  of  min(m,n)
               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(1) H(2) . . . H(k), 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:m) is stored on  exit  in
       A(i+1:m,i), and tau in TAU(i).


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