##  [Fixed-Point Iteration](/fixed-point-iteration-0) 

 Definition

An iterative procedure that seeks a fixed point x = G(x) of a mapping G by repeatedly applying G to an initial guess: x_{k+1} = G(x_k). Convergence depends on contractivity or related properties of G.

 

 

 

 

 

 





## Principle

Principle

The organizing idea is to recast the problem as finding a self-consistent point of a map and then use repeated application of that map; Banach fixed-point theorem gives a simple sufficient condition (contraction) guaranteeing unique fixed point and linear convergence.

 

 

 

 

 





## Demonstration

Demonstration

Solve x = cos(x) by iterating x_{k+1} = cos(x_k) starting from x_0; because cos is a contraction on [0,1], the iterates converge to the unique fixed point approximately 0.739085, illustrating simple Picard iteration for scalar nonlinear equations.

 

 

 

 

## Misapplication

Misapplication

Applying naive fixed-point iteration to a map with Lipschitz constant ≥1 or without proper preconditioning can fail to converge or converge extremely slowly; choosing a poor reformulation G(x) of f(x)=0 may prevent any progress.

 

 

 

 

 





## Consequence

Consequence

When the mapping is contractive or suitably damped, fixed-point iteration provides a simple and robust solver with predictable linear convergence and low per-iteration cost; it underlies many iterative schemes including Picard linearization for PDEs.

 

 

 

 

## Reversal

Reversal

The contrast is to Newton-type linearization: instead of repeatedly applying the original map, Newton solves linearized corrections producing potentially faster (superlinear or quadratic) local convergence at the cost of solving linear systems.

 

 

 

 

 





## Boundary

Boundary

Applicable when one can construct a map G with fixed points equivalent to the original problem and with contractive or averaged properties; excludes mappings that are not continuous or problems where only derivative-based rapid convergence is acceptable.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with Newton and quasi-Newton methods: fixed-point iteration is cheaper and simpler but slower; there is also tension with accelerated or multistep fixed-point variants that add memory or mixing to improve convergence.

 

 

 

 

 





## Synthesis

Synthesis

Fixed-point iteration reframes a problem as x = G(x) and repeatedly applies G, relying on contractivity or damping for convergence; it is conceptually simple, low-cost per iteration, and a foundation for many linearization and splitting algorithms, but its speed depends critically on the map's properties.