The asymptotically fastest algorithms for many linear algebra problems on integer matrices, including solving a system of linear equations and computing the determinant, use high-order lifting. Currently, high-order lifting requires the use of a randomized shifted number system to detect and avoid error-producing carries. By interleaving quadratic and linear lifting, we devise a new algorithm for high-order lifting that allows us to work in the usual symmetric range modulo p, thus avoiding randomization. As an application, we give a deterministic algorithm to assay if an n × n integer matrix A is unimodular. The cost of the algorithm is O((log n)n ω M(log n + log ||A||)) bit operations, where ||A|| denotes the largest entry in absolute value, and M(t) is the cost of multiplying two integers bounded in bit length by t.
We present a new heuristic algorithm for computing the determinant of a nonsingular n × n integer matrix. Extensive empirical results from a highly optimized implementation show the running time grows approximately as n 3 log n, even for input matrices with a highly nontrivial Smith invariant structure. We extend the algorithm to compute the Hermite form of the input matrix. Both the determinant and Hermite form algorithm certify correctness of the computed results.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.