Iteração de Ponto Fixo (1 passo)

The Fixed Point Iteration calculator solves equations of the form f(x)=0 by rewriting them as x = g(x) and applying the iteration x_{n+1} = g(x_n). Starting from an initial guess x0, the generated sequence converges to a root if the function g is suitable and the initial point is close enough. The method is a fundamental numerical analysis technique and serves as the basis for more advanced methods like Newton-Raphson.

The operation is simple: enter the function g(x), the initial value x0, and the number of iterations desired. The calculator repeatedly applies g to the last obtained value, displaying each step. It is important that g be a contraction mapping (derivative less than 1 in magnitude) near the root to ensure convergence. Otherwise, the sequence may diverge or oscillate.

Use this calculator to find roots of nonlinear equations, especially when other methods are complex. It is useful in numerical analysis courses, physics, and engineering. For example, to solve x = cos(x), simply set g(x)=cos(x) and iterate. The method is also used in fixed point problems in economics and dynamical systems.

Caution: choose an appropriate function g; not every rewriting of f(x)=0 into x=g(x) converges. Check that the derivative of g at the fixed point is less than 1. Also, the initial guess must be close to the root. The calculator does not automatically check convergence; monitor the values to see if they stabilize.