Selecting Algorithms for Black Box Matrices: Checking for Matrix Properties That Can Simplify Computations

TL;DR

This paper presents methods to automatically identify matrix properties of black box matrices, enabling the selection of more efficient algorithms by exploiting structural features and preconditioning techniques, especially in small-field cases.

Contribution

It introduces processes to check and certify matrix properties that facilitate the use of faster algorithms and reduce preconditioning, improving computational efficiency for black box matrices.

Findings

Methods to certify matrix sparsity and structural properties.

Identification of properties enabling superfast algorithms.

Reduction of preconditioning in small-field cases.

Abstract

Processes to automate the selection of appropriate algorithms for various matrix computations are described. In particular, processes to check for, and certify, various matrix properties of black box matrices are presented. These include sparsity patterns and structural properties that allow "superfast" algorithms to be used in place of black-box algorithms. Matrix properties that hold generically, and allow the use of matrix preconditioning to be reduced or eliminated, can also be checked for and certified - notably including in the small-field case, where this presently has the greatest impact on the efficiency of the computation.