Lecture

Convergence of Fixed Point Methods

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Description

This lecture covers the convergence of fixed point methods, discussing the global convergence property and uniqueness proofs. It explores the application of fixed point iterations and the consequences of Bolzano's theorem. The instructor presents the convergence criteria and the implications of different convergence rates, emphasizing the importance of the stopping criterion. The lecture also delves into the Newton method and its relation to fixed point iterations, highlighting the significance of local convergence and the modified Newton method.

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Ontological neighbourhood

Mathematics

Analysis: Differential anaysis

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X-ray tube

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Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

Rate of convergence

In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence that converges to is said to have order of convergence and rate of convergence if The rate of convergence is also called the asymptotic error constant. Note that this terminology is not standardized and some authors will use rate where this article uses order (e.g., ).

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.

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