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## slaorhr_col_getrfnp2()

recursive subroutine slaorhr_col_getrfnp2 | ( | integer | m,
|  | integer | n,
|  | real, dimension( lda, * ) | a,
|  | integer | lda,
|  | real, dimension( * ) | d,
|  | integer | info )

SLAORHR_COL_GETRFNP2

Download DLAORHR_GETRF2NP + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> SLAORHR_COL_GETRFNP2 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  Gaussian elimination is at
      !>    least one in absolute value (so that division-by-zero 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 recursive version of the LU factorization algorithm.
      !> Denote A - S by B. The algorithm divides the matrix B into four
      !> submatrices:
      !>
      !>        [  B11 | B12  ]  where B11 is n1 by n1,
      !>    B = [ -----|----- ]        B21 is (m-n1) by n1,
      !>        [  B21 | B22  ]        B12 is n1 by n2,
      !>                               B22 is (m-n1) by n2,
      !>                               with n1 = min(m,n)/2, n2 = n-n1.
      !>
      !>
      !> The subroutine calls itself to factor B11, solves for B21,
      !> solves for B12, updates B22, then calls itself to factor B22.
      !>
      !> For more details on the recursive LU algorithm, see [2].
      !>
      !> SLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
      !> routine SLAORHR_COL_GETRFNP, which uses blocked code calling
      !> Level 3 BLAS to update the submatrix. However, SLAORHR_COL_GETRFNP2
      !> is self-sufficient and can be used without SLAORHR_COL_GETRFNP.
      !>
      !> [1] ,
      !>     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
      !>     E. Solomonik, J. Parallel Distrib. Comput.,
      !>     vol. 85, pp. 3-31, 2015.
      !>
      !> [2] , F. Gustavson, IBM J. of Res. and Dev.,
      !>     vol. 41, no. 6, pp. 737-755, 1997.
      !>

Parameters

  [in]

  M

      !>          M is INTEGER
      !>          The number of rows of the matrix A.  M >= 0.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The number of columns of the matrix A.  N >= 0.
      !>

  [in,out]

  A

      !>          A is REAL array, dimension (LDA,N)
      !>          On entry, the M-by-N matrix to be factored.
      !>          On exit, the factors L and U from the factorization
      !>          A-S=L*U; the unit diagonal elements of L are not stored.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of the array A.  LDA >= max(1,M).
      !>

  [out]

  D

      !>          D is REAL array, dimension min(M,N)
      !>          The diagonal elements of the diagonal M-by-N sign matrix S,
      !>          D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
      !>          be only plus or minus one.
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit
      !>          < 0:  if INFO = -i, the i-th argument had an illegal value
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

Contributors:

      !>
      !> November 2019, Igor Kozachenko,
      !>                Computer Science Division,
      !>                University of California, Berkeley
      !>
      !>