p-Adic Stability In Linear Algebra

TL;DR

This paper investigates the p-adic stability of matrix operations using differential precision methods, showing lattice-based approaches outperform naive methods in accuracy and efficiency, with practical numerical validation.

Contribution

It introduces the application of differential precision to analyze p-adic stability of matrix operations, highlighting the superiority of lattice-based methods over naive approaches.

Findings

Lattice-based methods improve p-adic stability in matrix multiplication.

Differential methods provide better accuracy for determinants and LU factorization.

Numerical experiments confirm theoretical advantages of the proposed methods.

Abstract

Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants , characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.