SRC/dlaorh...getrfnp2.f(3)

💖 NAME 💖

SRC/dlaorhr_col_getrfnp2.f

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✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

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

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🎀 Function/Subroutine Documentation 🎀

recursive subroutine dlaorhr_col_getrfnp2 (integer m, integer n, double
    precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) d, integer info)

DLAORHR_COL_GETRFNP2

🎯 Purpose 🎯

Okay bestie, so 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 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.

References, because citations are iconic too: [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 🌸

• 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.

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👑 Author 👑

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