Fixed-point iteration

Objective. Rearrange equations and generate successive approximations.

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2. Worked example
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Learn: the key idea

Fixed-point iteration solves f(x) = 0 by rearranging it into the form x = g(x), then generating successive approximations with xₙ₊₁ = g(xₙ) from a starting value x₀. If the sequence converges, it approaches a root of the original equation. For x² − x − 2 = 0, rearranging gives x = √(x+2), and iterating from x₀ = 2 gives x₁ = 2, confirming convergence to the root x = 2. Always check the sequence is settling towards a consistent value before stating the root.

Worked example

Using fixed-point iteration with xₙ₊₁ = √(3xₙ + 1) and x₀ = 2, find x₂ to 3 decimal places.

1 $Compute x₁ = \sqrt(3(2)+1) = \sqrt7 = 2.6458$
2 $Round x₁ to 2.646 for the next step$
3 $Compute x₂ = \sqrt(3(2.646)+1) = \sqrt8.938$
4 $Evaluate \sqrt8.938 = 2.990$

Answer: x₂ = 2.990

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Common mistake. A common mistake is choosing a rearrangement for which the iteration diverges, then still reporting the diverging values as an approximate root. Always test a few iterations first and switch rearrangement if the values grow apart instead of converging.

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