Fixedpointiterationer

Fixedpointiterationer, commonly called fixed-point iteration, is a numerical method used to compute solutions to equations by transforming them into a fixed-point problem x = g(x). The objective is to find a fixed point x* of g, where applying g to x* yields x* itself.

The method is implemented by choosing an initial guess x0 and then generating a sequence x1, x2, … with xn+1 = g(xn). Convergence to a fixed point depends on the properties of g. A sufficient condition is that g is a contraction on some interval I, meaning there exists a constant L with 0 ≤ L < 1 such that |g(x) − g(y)| ≤ L|x − y| for all x, y in I. If this holds and the fixed point x* lies in I, the iteration converges to x* for any initial value x0 in I. The error typically decreases linearly, with |xn − x*| ≤ (L^n)/(1 − L) |x1 − x0|, and asymptotically |xn − x*| ≈ C L^n for some constant C.

A local convergence criterion is that, if g is differentiable near x* and |g′(x*)| < 1, the method converges when started sufficiently close to x*. Practical considerations include selecting a good initial guess, and stopping criteria such as |xn+1 − xn| < tol or |f(xn+1)| < tol, where the original equation is expressed as f(x) = 0.

Fixed-point iteration is widely used for solving nonlinear equations, often after reformulating the original problem into x = g(x). Its success hinges on the contraction property or acceptable local behavior of g near the desired solution.