Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos
Newton’s iterative method
References
Images
Newton's method
Britannica AI Icon

Newton’s iterative method

mathematics
Written by
The Information Architects of Encyclopaedia Britannica

The Information Architects maintain a master list of the topics included in the corpus of  Encyclopædia Britannica, and create and manage the relationships between them.
The Information Architects of Encyclopaedia Britannica
Last updated

Quick Summary

Newton's method
Newton's method Newton's method of iteration, in which each successive approximation is based on the preceding one and the slope of the tangent line.

Newton's method is a mathematical technique used to find approximate solutions to equations where a polynomial expression equals zero, meaning f(x) = 0. It was developed by Isaac Newton in 1669 and later simplified by Joseph Raphson.1

The method begins with an initial guess, x0. A tangent line is drawn to the graph of the function at this point, and where this line intersects the x-axis provides a new, improved guess, x1. This is calculated using the formula:

x1 = x0 - f(x0) / f'(x0)

Here, f'(x0) represents the derivative, or slope, of the function at x0. This iterative process is repeated, using the new guess to find the next, until the desired level of accuracy is achieved. A general formula for this process is:1

xn + 1 = xn - f(xn) / f'(xn).1

For example, to find the square root of 5 (solving x2 - 5 = 0), with an initial guess of 2, the method quickly converges to a highly accurate approximation.1

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
3 Britannica Sources
  1. Newton’s method
  2. Newton and Infinite Series
  3. analysis - Calculus flourishes

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.