# Iterative methods for linear systems of equations: A brief historical journey

## TL;DR

This paper provides a concise historical overview of iterative methods for solving linear systems, highlighting key developments from Gauss to the computer era, emphasizing major ideas shaping the field.

## Contribution

It offers a non-exhaustive historical survey emphasizing the evolution of iterative methods and their foundational ideas over time.

## Key findings

- Gauss developed the first iterative method.
- Significant progress in the 20th century for least-squares and PDE discretizations.
- Computers in the 1950s greatly accelerated development.

## Abstract

This paper presents a brief historical survey of iterative methods for solving linear systems of equations. The journey begins with Gauss who developed the first known method that can be termed iterative. The early 20th century saw good progress of these methods which were initially used to solve least-squares systems, and then linear systems arising from the discretization of partial different equations. Then iterative methods received a big impetus in the 1950s - partly because of the development of computers. The survey does not attempt to be exhaustive. Rather, the aim is to underline the way of thinking at a specific time and to highlight the major ideas that steered the field.