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

Bestie, this page is from the Library Functions Manual, and the page title is SRC/slaorh...getrfnp2.f(3). Buckle up; we're doing modified LU without pivoting. 🍓

💖 NAME 💖

SRC/slaorhr_col_getrfnp2.f

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• recursive subroutine slaorhr_col_getrfnp2 (m, n, a, lda, d, info) — literally the icon: SLAORHR_COL_GETRFNP2.

🧚 Function/Subroutine Documentation 🧚

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

SLAORHR_COL_GETRFNP2

🎯 Purpose

okay so SLAORHR_COL_GETRFNP2 computes the modified LU factorization without pivoting of a real general M-by-N matrix A. The factorization has this 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 🌈

• 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 166 of file slaorhr_col_getrfnp2.f.

🤖 Author

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