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Lecture# Stopping Criteria for Nonlinear Equations

Description

This lecture covers the stopping criteria for fixed point iteration in nonlinear equations, focusing on the convergence conditions and error control. It also discusses the stopping criteria for Newton's method, emphasizing the importance of the increment and error tolerance. The instructor explains the key concepts through examples and practical applications, highlighting the significance of accurate error estimation in iterative methods.

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Instructors (2)

Related concepts (31)

MATH-250: Numerical analysis

Construction et analyse de méthodes numériques pour la solution de problèmes d'approximation, d'algèbre linéaire et d'analyse

Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is which gives rise to the sequence of iterated function applications which is hoped to converge to a point . If is continuous, then one can prove that the obtained is a fixed point of , i.e., More generally, the function can be defined on any metric space with values in that same space.

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.

Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.

Fixed point (mathematics)

hatnote|1=Fixed points in mathematics are not to be confused with other uses of "fixed point", or stationary points where math|1=f(x) = 0. In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function. Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c.

Newton's method

In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then is a better approximation of the root than x0.

Related lectures (42)

Higher Order Methods: Iterative TechniquesMATH-251(a): Numerical analysis

Covers higher order methods for solving equations iteratively, including fixed point methods and Newton's method.

Nonlinear Equations: Fixed Point MethodMATH-250: Numerical analysis

Covers the topic of nonlinear equations and the fixed point method.

Numerical Analysis: Nonlinear EquationsMATH-251(c): Numerical analysis

Explores the numerical analysis of nonlinear equations, focusing on convergence criteria and methods like bisection and fixed-point iteration.

Convergence of Fixed Point Methods

Explores the convergence of fixed point methods and the implications of different convergence rates.

Newton's Method: ConvergenceMATH-250: Numerical analysis

Explores the convergence of Newton's method for solving nonlinear equations and the importance of selecting appropriate initial guesses.