DGERQF  -  compute  an  RQ  factorization of a real M-by-N
       matrix A


SYNOPSIS

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

           INTEGER        INFO, LDA, LWORK, M, N

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


PURPOSE

       DGERQF  computes  an  RQ  factorization  of  a real M-by-N
       matrix A: A = R * 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) DOUBLE PRECISION array, dimension
               (LDA,N)
               On  entry,  the M-by-N matrix A.  On exit, if m <=
               n, the upper triangle  of  the  subarray  A(1:m,n-
               m+1:n) contains the M-by-M upper triangular matrix
               R; if m >= n, the elements on and  above  the  (m-
               n)-th  subdiagonal contain the M-by-N upper trape­
               zoidal matrix R; the remaining elements, 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,M).   For optimum performance LWORK >= M*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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is  stored
       on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).


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