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Fixed Point Iteration

Solves equations by repeatedly applying a function until the value stops changing.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Numerical Methods
Difficulty: Beginner
Family: Fixed Point Method
Tier: Extended
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 2
Code example: Not ready
Visualization: Ready

Overview

Fixed Point Iteration rewrites a problem as x equals g(x) and repeatedly applies g to approach a stable value.

Core Concepts

numeric approximationiterative updateroot estimatefixed point methodconvergence checkerror control

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(1)

Average

O(t)

Worst

O(t)

Space

O(1)

When to Use

Use Fixed Point Iteration when you need to apply an update function repeatedly.
Use it for simple nonlinear solves with inputs like update function.
Use it when an approximate numeric result is acceptable and the input assumptions are satisfied.
Use it when the Fixed Point Method tradeoff is acceptable for the problem size.

When Not to Use

Avoid it when rounding error, ill-conditioning, or convergence failure would hide the real answer.
Avoid it when the required output is not approximate fixed point.

What It Does

Computes fixed point method within Fixed Point Method workloads (O(t) time, O(1) space).
Works by Solves equations by repeatedly applying a function until the value stops changing.
Maintains key invariants across numeric approximation, iterative update, root estimate.

Use Cases

Simple nonlinear solves
Iterative numerical methods
Teaching convergence

Inputs

Update function
Initial estimate
Tolerance

Outputs

Approximate fixed point

Key Formulas (1)

Fixed-point update

x_next = g(x)

x_next = g(x)

Repeats an update function until the value stops changing meaningfully.

Pseudocode

1.Choose an initial estimate
2.Apply the update function
3.Compare the new and old values
4.Repeat until the change is small
5.Return the stable estimate

Canonical Example

Input: equation = x^2 - 9 = 0 or matrix system Ax = b
Output: approximate numeric solution
Explanation: Fixed Point Iteration: The method iteratively or algebraically approximates the requested mathematical result.

Implementation Notes

Keep the input contract explicit: Update function; Initial estimate; Tolerance.
Use this step as the implementation spine: Choose an initial estimate.
Then preserve the next state change: Apply the update function.
Use small examples with known answers to verify the numeric update before scaling up.

Common Pitfalls

Convergence depends on the chosen rearrangement of the equation.
Stopping based only on iteration count instead of checking residual, tolerance, or error.
Returning approximate fixed point without handling empty, invalid, or boundary inputs.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show repeated function applications approaching a stable value.

Step 1 of 5: Choose an initial estimate

fixed-point-steps

Notes

Fixed Point Iteration operates in O(t) time and O(1) space within the Fixed Point Method family. Key governing relation: x_next = g(x). Often compared with Newton's Method and Secant Method.