3.6.2—Iterative approximations

With x_{n+1}=g(x_n), a fixed point α satisfies g(α)=α. Starting from x₀ creates a sequence intended to approach α.

Choose a valid starting value, retain guard digits, and stop when successive values or the residual meet the tolerance. Verify the final value in f(x)=0.

For x=cosx, x₀=1 gives iterates that approach approximately 0.739; reporting the last iterate without a tolerance is incomplete.

Iteration can oscillate or diverge even when a root exists; apparent agreement of early digits is not a proof.