Matrices with Tunable Infinity-Norm Condition Number and No Need for Pivoting in LU Factorization

We propose a two-parameter family of nonsymmetric dense n× n matrices A(α,β) for which LU factorization without pivoting is numerically stable, and we show how to choose α and β to achieve any value of the ∞-norm condition number. The matrix A(α,β) can be formed from a simple formula in O(n²) flops. The matrix is suitable for use in the HPL-AI Mixed-Precision Benchmark, which requires an extreme scale test matrix (dimension n>10⁷) 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.