What is a fixed point?

A fixed point of g(x) is a value x* such that g(x*) = x*. Finding this value solves the equation x = g(x), which is equivalent to f(x) = x - g(x) = 0.

When does the method converge?

The iteration converges if |g'(x)| = 1, the sequence will diverge. Try rewriting g(x) or choosing a better rearrangement of your equation.

What is the classic cos(x) example?

Setting g(x) = cos(x) and starting from x0 = 1 converges to the Dottie number ~0.7390851, the unique fixed point of cosine. |g'(x)| = |sin(x)| < 1 there, ensuring convergence.

What expressions can I use in g(x)?

This tool uses math.js, so you can use standard functions: sin, cos, tan, exp, log, sqrt, abs, pow, pi, e, and arithmetic operators + - * / ^. Example: sqrt(x + 2) or (x^2 + 3) / 4.

How do I choose the initial guess x0?

Start with a value close to the expected root. You can estimate it from a graph of y = x and y = g(x) — the fixed point is where they intersect.

What does "max iterations" control?

It caps the table size to prevent infinite loops when the method diverges. If convergence has not been reached by the maximum count, the tool reports "Did not converge."

Fixed Point Iteration

Solve x = g(x) numerically: enter your g(x) expression, initial guess, tolerance, and see every iteration in a table until convergence.

Parameters

g(x) expression (x = g(x) form)

Initial guess x₀

Tolerance

Max iterations

Quick examples

Enter parameters and click Run Iteration

n	x_n	g(x_n)	Error \|x_n+1 − x_n\|

Summary

The fixed-point iteration method solves equations of the form x = g(x) by repeatedly applying the function g starting from an initial guess x0. Each new approximation is x_(n+1) = g(x_n), and the process continues until the change between successive values falls below a chosen tolerance. This tool evaluates any g(x) expression you enter, displays the full iteration table, and reports whether the sequence converged or diverged.

How it works

Express your equation in the form x = g(x) and enter g(x) in the input field.
Provide an initial guess x0 close to the expected root.
Set a convergence tolerance (e.g. 0.0001) and a maximum iteration count.
Click "Run Iteration" to compute x_(n+1) = g(x_n) repeatedly.
Each step is displayed in a table showing n, x_n, g(x_n), and the absolute error |x_(n+1) - x_n|.
Iteration stops when the error falls below the tolerance (converged) or the maximum count is reached (diverged or slow convergence).

Use cases

Find roots of transcendental equations like cos(x) = x.
Verify fixed-point iteration convergence in numerical analysis courses.
Explore how the choice of g(x) and x0 affects convergence speed.
Demonstrate divergence when |g'(x)| >= 1 near the root.
Solve engineering equations iteratively without needing derivatives.
Cross-check hand-computed iteration tables for homework problems.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu