slaorhr_col_getrfnp.f - Man Page

subroutine slaorhr_col_getrfnp (m, n, a, lda, d, info)
SLAORHR_COL_GETRFNP

SLAORHR_COL_GETRFNP  

Purpose:

 SLAORHR_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.

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 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.

LDA

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

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.

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 145 of file slaorhr_col_getrfnp.f.