# Who discovered the fixed point iteration method?

I was working on the area of fixed point iteration. I couldn't find any historical account regarding the subject. Who was the first to use the this method?

• I suppose another interesting point in the historical development of the method is who first described conditions under which the iteration converges. – Michael E2 Dec 24 '16 at 17:55

## 1 Answer

There are several "fixpoint iteration methods" and several fixed point theorems underlying them. One of the earliest uses was "Picard's iteration method" for proving existence of solutions of ODE. It is based on the "Banach fixed point theorem", though Banach was not born yet when Picard discovered it.

Edit. Thanks to those who recalled Newton's method, which was discovered in a special case in Babylon thousands years ago. There are indeed very many "fixpoint methods".

• Earlier than Picard: "Newton's method" for solving certain algebraic equations. – Gerald Edgar Dec 22 '16 at 16:25
• The Wikipedia page linked to in the question contains a link to Babylonian method" of computing square roots (a special case of Newton's method), first described by Hero of Alexandria. But this historical information is something that the OP could find out themselves if only they chose to follow the links provided. – Margaret Friedland Dec 22 '16 at 17:21