2.6.3—Iteration formulae

For x_{n+1}=g(x_n), convergence near a fixed point is encouraged when |g′(x)|<1 there; the derivative also indicates how quickly errors shrink locally.

Derive g from the original equation without changing the root set, select an admissible initial value, and verify the final approximation in the original function.

Rearranging x³+x−1=0 as x=(1−x)^{1/3} may behave differently from another rearrangement; compare the local slope before iterating.

An iteration formula is not automatically valid just because it is algebraically equivalent; convergence is a separate question.