3.6.3—Iteration formulae

For x_{n+1}=g(x_n), a fixed point is locally attractive when |g′(α)|<1; a smaller magnitude generally means faster local error reduction.

Derive g from the original equation, check the derivative on the intended interval and compare stopping errors with the function residual.

Two algebraically equivalent rearrangements of x³+x−1=0 can have different g′ near the root; choose the one with smaller local magnitude.

Algebraic equivalence of equations does not imply numerical equivalence of iterations.