zgetrf.f(3)

🌸 NAME 🌸

Okay so bestie, this is zgetrf.f(3), and the manual name is LAPACK — we’re folding that in right here so you don’t miss it. This page is all about zgetrf.f, the ZGETRF variant that’s an iterative version of Sivan Toledo’s recursive LU algorithm. Slay. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine zgetrf (M, N, A, LDA, IPIV, INFO) — ZGETRF VARIANT: iterative version of Sivan Toledo’s recursive LU algorithm. Periodt.

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💖 Function/Subroutine Documentation 💖

subroutine zgetrf (integer M, integer N, complex*16, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) IPIV, integer INFO)

ZGETRF VARIANT: iterative version of Sivan Toledo’s recursive LU algorithm.

Purpose:

ZGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. Iconic behavior, honestly.

The factorization has the form

A = P * L * U

where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

This code implements an iterative version of Sivan Toledo’s recursive LU algorithm[1]. For square matrices, this iterative versions should be within a factor of two of the optimum number of memory transfers.

The pattern is as follows, with the large blocks of U being updated in one call to DTRSM, and the dotted lines denoting sections that have had all pending permutations applied:

         1 2 3 4 5 6 7 8
        +-+-+---+-------+------
        | |1|   |       |
        |.+-+ 2 |       |
        | | |   |       |
        |.|.+-+-+   4   |
        | | | |1|       |
        | | |.+-+       |
        | | | | |       |
        |.|.|.|.+-+-+---+  8
        | | | | | |1|   |
        | | | | |.+-+ 2 |
        | | | | | | |   |
        | | | | |.|.+-+-+
        | | | | | | | |1|
        | | | | | | |.+-+
        | | | | | | | | |
        |.|.|.|.|.|.|.|.+-----
        | | | | | | | | |

The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in the binary expansion of the current column. Each Schur update is applied as soon as the necessary portion of U is available. Obsessed with the bit lore, ngl. 🧚

[1] Toledo, S. 1997. Locality of Reference in LU Decomposition with Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997), 1065-1081. http://dx.doi.org/10.1137/S0895479896297744

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 COMPLEX*16 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 = P*L*U; the unit diagonal elements of L are not stored. No cap. 🫶
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• IPIV — IPIV is INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
• INFO — INFO is INTEGER.
  • = 0: successful exit 💖
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 🚨 Bestie, that’s the serious one — do not ignore it.

Author:

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

Date:

December 2016

Definition at line 136 of file VARIANTS/lu/REC/zgetrf.f.

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🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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Byeeee bestie 💋 this page is from LAPACK Version 3.8.0, dated Tue Nov 14 2017, and it’s zgetrf.f(3) — mwah. 🎀🦋