Computational Complexity and Numerical Stability of Linear Problems

TL;DR

This paper surveys the interplay between computational complexity and numerical stability in linear algebra, emphasizing matrix multiplication's role and analyzing classical and recent developments in the field.

Contribution

It provides a comprehensive overview of algebraic complexity theory and error analysis in numerical linear algebra, highlighting the importance of matrix multiplication.

Findings

Matrix multiplication complexity impacts many linear algebra algorithms.

Numerical stability is crucial for reliable results in linear computations.

Historical and modern approaches to matrix multiplication are discussed.

Abstract

We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the algebraic complexity theory as well as the general error analysis for matrix multiplication and related problems. We emphasize the central role played by the matrix multiplication problem and discuss historical and modern approaches to its solution.