2.6.2—Iterative approximations

Iteration replaces x by x_{n+1}=g(x_n). A fixed point α satisfies g(α)=α, corresponding to a root of the rearranged equation.

Choose a starting value in the stated interval, compute enough figures during iteration, and stop using a tolerance on successive values or the residual. Different rearrangements can converge differently.

For x=cos x, starting x₀=0 gives a sequence approaching about 0.739; starting values should remain in a region where g behaves stably.

A few stable-looking digits do not prove convergence, and iteration can diverge or enter a cycle even when the equation has a root.