LAORHR_COL_GETRFNP - 5.2 English - 68552

   LAORHR_COL_GETRFNP computes the modified LU factorization without
   pivoting of a real general M-by-N matrix A. The factorization has
   the form:

       A - S = L * U,

   where:
      S is a m-by-n diagonal sign matrix with the diagonal D, so that
      D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
      as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
      i-1 steps of Gaussian elimination. This means that the diagonal
      element at each step of "modified" Gaussian elimination is
      at least one in absolute value (so that division-by-zero not
      not possible during the division by the diagonal element);

      L is a M-by-N lower triangular matrix with unit diagonal elements
      (lower trapezoidal if M > N);

      and U is a M-by-N upper triangular matrix
      (upper trapezoidal if M < N).

   This routine is an auxiliary routine used in the Householder
   reconstruction routine SORHR_COL. In SORHR_COL, this routine is
   applied to an M-by-N matrix A with orthonormal columns, where each
   element is bounded by one in absolute value. With the choice of
   the matrix S above, one can show that the diagonal element at each
   step of Gaussian elimination is the largest (in absolute value) in
   the column on or below the diagonal, so that no pivoting is required
   for numerical stability [1].

   For more details on the Householder reconstruction algorithm,
   including the modified LU factorization, see [1].

   This is the blocked right-looking version of the algorithm,
   calling Level 3 BLAS to update the submatrix. To factorize a block,
   this routine calls the recursive routine SLAORHR_COL_GETRFNP2.

   [1] "Reconstructing Householder vectors from tall-skinny QR",
       G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
       E. Solomonik, J. Parallel Distrib. Comput.,
       vol. 85, pp. 3-31, 2015.