Fixed-Point Iteration Calculator

When to use

• Use fixed-point iteration when an equation can be rearranged as x=g(x) with a contractive mapping near the solution.
• It is useful for comparing how algebraically equivalent rearrangements change convergence.
• Use a safeguarded root finder when no stable mapping can be justified.

How to use

1. Enter the mapping g(x), not the original zero-form f(x).
2. Choose an initial value inside a region mapped back into itself.
3. Run the iteration and inspect x_n, g(x_n), update size, and the sequence visual.
4. Try another valid mapping or starting value when updates oscillate or grow.

Inputs

Iteration function g(x)

The mapping repeatedly evaluated in x_(n+1)=g(x_n).

Initial value x0

The first iterate, preferably inside an invariant interval.

Tolerance

The threshold for iterate change and fixed-point residual |g(x)-x|.

Maximum iterations

The permitted number of mapping evaluations.

Notation

Interpretation

• A fixed point solves the original equation only when the rearrangement is domain-valid.
• Shrinking updates indicate local convergence; alternating or growing values indicate instability.
• Two rearrangements of one equation can have different convergence behavior.

Convergence and accuracy guidance

• Local convergence is commonly associated with |g'(x)| below one near the fixed point.
• Compare |g(x_n)-x_n| with update size to detect rounded stagnation.
• Confirm that every iterate remains in the mathematical and physical domain.

Worked examples

Assumptions

• The mapping is equivalent to the intended equation over the working domain.
• g maps the working interval into itself and is locally contractive.
• All iterates remain finite in real arithmetic.

Practical guidance

• Derive alternative mappings before selecting one.
• Plot y=g(x) with y=x to inspect fixed points.
• Verify the final value in the original equation.

Common mistakes

• Using a rearrangement that is not contractive.
• Confusing small updates with a correct physical model.
• Forgetting to enter f(x) when residual checking is desired.

Limitations

• Convergence depends on g(x), not only on f(x).
• Oscillation or divergence can occur.
• The tool does not automatically find the best rearrangement.

Related workflow links