dlaorhr_col_getrfnp2.f - Man Page

recursive subroutine dlaorhr_col_getrfnp2 (m, n, a, lda, d, info)
DLAORHR_COL_GETRFNP2

DLAORHR_COL_GETRFNP2  

Purpose:

 DLAORHR_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 'modified' 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 DORHR_COL. In DORHR_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].

 DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
 routine DLAORHR_COL_GETRFNP, which uses blocked code calling
 Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
 is self-sufficient and can be used without DLAORHR_COL_GETRFNP.

 [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.

 [2] 'Recursion leads to automatic variable blocking for dense linear
     algebra algorithms', F. Gustavson, IBM J. of Res. and Dev.,
     vol. 41, no. 6, pp. 737-755, 1997.

Parameters

M

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

N

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

A

          A is DOUBLE PRECISION 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.

LDA

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

D

          D is DOUBLE PRECISION 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.

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

Definition at line 166 of file dlaorhr_col_getrfnp2.f.