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


SYNOPSIS

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

           INTEGER        INFO, LDA, LWORK, M, N

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


PURPOSE

       SGERQF 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) REAL 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) 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(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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