LU factorization with panel rank revealing pivoting and its communication avoiding version

TL;DR

This paper introduces LU_PRRP, a more stable LU factorization method using rank revealing QR, and CALU_PRRP, a communication-efficient, stable variant resistant to pathological cases, outperforming traditional methods.

Contribution

The paper proposes LU_PRRP and CALU_PRRP algorithms that improve stability and communication efficiency over existing LU factorization methods.

Findings

LU_PRRP is as stable as GEPP in practice.

CALU_PRRP reduces communication and is more stable than CALU.

Both methods resist pathological matrices like Wilkinson and Foster.

Abstract

We present the LU decomposition with panel rank revealing pivoting (LU_PRRP), an LU factorization algorithm based on strong rank revealing QR panel factorization. LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP). Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to pathological cases and easily solves the Wilkinson matrix and the Foster matrix. We also present CALU_PRRP, a communication avoiding version of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament pivoting, with the selection of the pivots at each step of the tournament being performed via strong rank revealing QR factorization. CALU_PRRP is more stable than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more stable in practice and is resistant to pathological cases on which GEPP and CALU fail.