Matrices with Tunable Infinity-Norm Condition Number and No Need for Pivoting in LU Factorization
Massimiliano Fasi, Nicholas John Higham · SIAM Journal on Matrix Analysis and Applications · 2021
We propose a two-parameter family of nonsymmetric dense $n\times n$ matrices $A(\alpha,\beta)$ for which LU factorization without pivoting is numerically stable, and we show how to choose $\alpha$ and $\beta$ to achieve any value of the $\infty$-norm condition number. The matrix $A(\alpha,\beta)$ can be formed from a simple formula in $O(n^2)$ flops. The matrix is suitable for use in the HPL-AI Mixed-Precision Benchmark, which requires an extreme scale test matrix (dimension $n>10^7$) that has a controlled condition number and can be safely used in LU factorization without pivoting. It is also of interest as a general-purpose test matrix.