# LU factorization with errors *

## TL;DR

This paper introduces new algorithms for detecting and correcting errors in LU factorization and triangular system solutions over any field, with efficiency depending on the number of errors, applicable to general linear system solving.

## Contribution

The paper presents error detection and correction algorithms for LU factorization that operate without extra information and adapt their complexity based on error count.

## Key findings

- Algorithms run in softly linear time relative to matrix size and errors
- Effective correction of errors in LU factorization over arbitrary fields
- Applications demonstrated in solving general linear systems

## Abstract

We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the dimension times the number of errors when there are few errors, smoothly growing to the cost of fast matrix multiplication as the number of errors increases. We also present applications to general linear system solving.