Fixed point

A fixed point of a transformation is an input that the transformation maps to itself. It is also called an invariant point.

Fixed-point iteration: A numerical method that repeatedly applies a function to approximate a point unchanged by that function. It searches for a fixed point by repeatedly feeding each output back as the next input.
Function (mathematics): A relation that assigns each input in a domain exactly one output in a codomain. A fixed point is defined by comparing an input with its function's output.
Root-finding algorithm: A numerical procedure for finding an input at which a specified function equals zero. An equation f(x)=0 can be rewritten as a fixed-point equation g(x)=x.
Periodic point: A point that returns to itself after a finite number of iterations of a transformation. A fixed point returns after one application; a periodic point may require several.
Banach fixed-point theorem: A theorem stating that a contraction on a complete metric space has a unique fixed point. It guarantees both a unique fixed point and convergence of iteration under precise conditions.
Iteration: The repeated application of a function or procedure to the result of its previous application. Iterating a map reveals whether points approach or move away from its fixed points.
Newton's method: An iterative method that estimates a root by using the tangent line at each current estimate. Its root estimates are fixed points of the iteration function used by the method.
Preimage: An input whose image under a function is a specified output. A preimage need only map to the point; a fixed point must be that point itself.
Brouwer fixed-point theorem: A theorem stating that every continuous map from a compact convex subset of Euclidean space to itself has a fixed point. It establishes existence even when no direct method for locating the fixed point is known.
Metric space: A set equipped with a distance function satisfying the axioms of a metric. Distance lets contraction conditions and convergence to fixed points be stated precisely.

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