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Lecture# Newton's Method: Convergence and Criteria

Description

This lecture covers the topic of non-linear equations, focusing on the Newton method and its convergence properties. It explains the iterative process of finding roots using Newton's method, discussing the criteria for convergence and stopping conditions. The lecture also delves into the interpretation of the method as a fixed-point method and explores the implications of different starting points on convergence.

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Related concepts (140)

Instructors (2)

MATH-251(c): Numerical analysis

Le cours présente des méthodes numériques pour la résolution de problèmes mathématiques comme des systèmes d'équations linéaires ou non linéaires, approximation de fonctions, intégration et dérivation

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.

Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.

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.

Related lectures (478)

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.

Numerical Analysis: Newton's MethodMATH-251(c): Numerical analysis

Explores Newton's method for finding roots of nonlinear equations and its interpretation as a second-order method.

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.

Method of Undetermined CoefficientsMATH-106(e): Analysis II

Explores the method of undetermined coefficients for solving non-homogeneous linear differential equations with constant coefficients.