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In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
Fixed Point Iterations. Given an equation of one variable, f(x) = 0, we use fixed point iterations as follows: Convert the equation to the form x = g(x). Start with an initial guess x0 ≈ r, where r is the actual solution (root) of the equation. Iterate, using xn+1 := g(xn) for n = 0, 1, 2, . . . . How well does this process work? We claim that:
Fixed point iteration shows that evaluations of the function g can be used to try to locate a fixed point. This is our first example of an iterative algortihm. The idea is to generate not a single answer but a sequence of values that one hopes will converge to the correct result.
Fixed point iteration is both a useful analytical tool, and a powerful algorithm. We will use fixed point iteration to learn about analysis and performance of algorithms, we will cover different implementations and their advantages and disadvantages, and we will look into several basic examples.
Fixed-point equations. A variant of stating equations as root-finding (\ (f (x) = 0\)) is fixed-point form: given a function \ (g:\mathbb {R} \to \mathbb {R}\) or \ (g:\mathbb {C} \to \mathbb {C}\) (or even \ (g:\mathbb {R}^n \to \mathbb {R}^n\); a later topic), find a fixed point of \ (g\).
Theorem 2.2 is a sufficient condition for a unique fixed point, i.e. if 2.2 is satisfied, fixed point is unique. BUT, 2.2 is not necessary, i.e. If 2.2 is not satisfied, a unique fixed point is still possible (see previous example and next example).
7 paź 2024 · Despite its simplicity, fixed-point iteration underpins many modern solvers used in numerical analysis to solve linear and nonlinear systems of equations. In this third instalment of the...
