DLAORHR_COL_GETRFNP.F(3)

hey bestie 💖 this is SRC/dlaorhr_col_getrfnp.f(3) from the Library Functions Manual, and we're about to get into the DLAORHR_COL_GETRFNP of it all. no cap, she's a LAPACK girlie doing modified LU factorization things. let's go 🫶

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🌸 NAME 🌸

SRC/dlaorhr_col_getrfnp.f — that's the name on the page, bestie. the routine it documents is DLAORHR_COL_GETRFNP. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

subroutine dlaorhr_col_getrfnp (m, n, a, lda, d, info)

DLAORHR_COL_GETRFNP

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💖 FUNCTION/SUBROUTINE DOCUMENTATION 💖

here's the full signature, literally exactly:

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

DLAORHR_COL_GETRFNP

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🎀 Purpose 🎀

okay so DLAORHR_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 Gaussian elimination is at least one in absolute value (so that division-by-zero is 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 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 DLAORHR_COL_GETRFNP2. 👑

[1] G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

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🧚 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.

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👑 Author 👑

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

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🫶 Contributors 🫶

November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley. 🎀

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Definition at line 145 of file dlaorhr_col_getrfnp.f.

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💀 AUTHOR 💀

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole vibe, bestie 💋 this page is from LAPACK, Version 3.12.0, with footer tag SRC/dlaorh...l_getrfnp.f(3) — mwah, bye 🎀✨