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 '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 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] '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.

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

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 November 2019, Igor Kozachenko,


                Computer Science Division,


                University of California, Berkeley