Iteration
Use a recurrence xn+1 = f(xn) to approximate solutions. Higher tier.
Higher-tier numerical method. Rearrange an equation into x = f(x) form. Start with an initial guess x0; compute x1 = f(x0), x2 = f(x1), and so on. Successive values converge (or diverge) to a solution.
Worked examples
x² − x − 5 = 0 rearranged: x = √(x + 5). Starting x0 = 2: x1 = √7 ≈ 2.65; x2 ≈ 2.77; x3 ≈ 2.79; converging to ~2.79.
Used when an equation can't be solved analytically.
Calculator: enter x0, then iterate by replacing x with the previous answer.
Frequently asked questions
Always converges?
No — some rearrangements diverge. Try a different rearrangement if so.
How many iterations?
Until consecutive values agree to the required precision (e.g. 4 dp).