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


SYNOPSIS

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

           INTEGER        INFO, LDA, LWORK, M, N

           DOUBLE         PRECISION A( LDA, * ), TAU( * ),  WORK(
                          * )


PURPOSE

       DGEQRF 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
               (min(M,N))
               The  scalar  factors  of the elementary reflectors
               (see Further Details).

       WORK     (workspace/output)  DOUBLE  PRECISION  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  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 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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