FixpunktIterationen

Fixpunktiterationen, also known as fixed-point iteration, is a numerical method used to find solutions to equations of the form x = g(x), where g is a given function. The method is based on the idea of iteratively applying the function g to an initial guess x₀, producing a sequence of approximations: x_{n+1} = g(x_n). Under suitable conditions, this sequence converges to a fixed point x*, satisfying x* = g(x*).

The convergence of fixpunktiterationen depends on the properties of g and the choice of the initial approximation. It generally requires g to be continuous and, more strongly, to have a Lipschitz constant less than 1 in a neighborhood of the fixed point. If these conditions are met, the iteration converges linearly to the fixed point.

This method is widely used due to its simplicity and applicability to a broad class of problems, including solving nonlinear equations and systems of equations. However, it has limitations; the convergence can be slow or may not occur if the function does not satisfy the necessary conditions.

Enhancements to the basic fixpunktiterationen include under-relaxation and over-relaxation techniques, which modify the iterative step to improve convergence behavior. The method is closely related to other numerical techniques such as Newton's method, which may converge faster but require derivatives.

In summary, fixpunktiterationen is a fundamental iterative method for solving equations, valued for its conceptual simplicity and versatility, with convergence heavily influenced by the properties of the function involved and the initial guess.