Müller’s Method

Abstract

(p 0 , f (p 0)) (p 2 , f(p 2)) p 0 t = h 0 h 1 h 0 p 1 t = h 1 p 2 t = 0 p 3 t = z y = f(x) Figure 2.17 The starting approximations p 0 , p 1 , and p 2 for Muller's method, and the differences h 0 and h 1. Muller's method is a generalization of the secant method, in the sense that it does not require the derivative of the function. It is an iterative method that requires three starting points (p 0 , f (p 0)), (p 1 , f (p 1)), and (p 2 , f (p 2)). A parabola is constructed that passes through the three points; then the quadratic formula is used to find a root of the quadratic for the next approximation. It has been proved that near a simple root Muller's method converges faster than the secant method and almost as fast as Newton's method. The method can be used to find real or complex zeros of a function and can be programmed to use complex arithmetic. Without loss of generality, we assume that p 2 is the best approximation to the root and consider the parabola through the three starting values, shown in Figure 2.17. Make the change of variable (9) t = x − p 2 , and use the differences (10) h 0 = p 0 − p 2 and h 1 = p 1 − p 2. Consider the quadratic polynomial involving the variable t: (11) y = at 2 + bt + c.