Editing Iterative method (section)

==Methods of successive approximation==
Mathematical methods relating to successive approximation include:
* [[Babylonian method]], for finding square roots of numbers<ref>{{Cite web |date=December 1, 2000 |title=Babylonian mathematics |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_mathematics/ |access-date= |website=Babylonian mathematics}}</ref>
* [[Fixed-point iteration]]<ref>{{Cite book |last=day |first=Mahlon |title=Fixed-point theorems for compact convex sets |date=November 2, 1960 |publisher=Mahlon M day |language=en}}</ref>
*  Means of finding zeros of functions:
** [[Halley's method]]
** [[Newton's method]]
* Differential-equation matters:
** [[Picard–Lindelöf theorem]], on existence of solutions of differential equations
** [[Runge–Kutta methods]], for numerical solution of differential equations

===History===
[[Jamshīd al-Kāshī]] used iterative methods to calculate the sine of  1° and {{pi}} in ''The Treatise of Chord and Sine'' to high precision. 
An early iterative method for [[Gauss–Seidel method|solving a linear system]] appeared in a letter of [[Carl Friedrich Gauss|Gauss]] to a student of his.  He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest {{citation needed|date=December 2019}}.

The theory of stationary iterative methods was solidly established with the work of [[D.M. Young]] starting in the 1950s. The conjugate gradient method was also invented in the 1950s, with independent developments by [[Cornelius Lanczos]], [[Magnus Hestenes]] and [[Eduard Stiefel]], but its nature and applicability were misunderstood at the time. Only in the 1970s was it realized that conjugacy based methods work very well for [[partial differential equation]]s, especially the elliptic type.