Successive Approximation Root Finding
The Fixed Point Method (also known as Successive Approximation) is an iterative technique that finds roots by rearranging the equation f(x) = 0 into the form x = g(x), then using the iteration xi+1 = g(xi). The method converges to a fixed point where x = g(x).
Convergence Condition: The method converges if |g'(x)| < 1 near the fixed point. Choose g(x) carefully to ensure convergence.
Original equation: x³ - x - 1 = 0
Rearranged to: x = ∛(x + 1) or x = (x³ - 1)/x + x or x = 1/(x² - 1) + x
Choose g(x): g(x) = (x + 1)^(1/3) for better convergence