Muller Method

Complex Zeros: Müller’s Method

One problem with applying the Secant, False Position, or Newton’s method to polynomials is the possibility of the polynomial having complex roots even when all the coefficients are real numbers. If the initial approximation is a real number, all subsequent approximations will also be real numbers. One way to overcome this difficulty is to begin with a complex initial approximation and do all the computations using complex arithmetic. An alternative approach has its basis in the following theorem.
Theorem 2: If z=a+bi is a complex zero of multiplicity m of the polynomial P(x) with real coefficients, then ¯z=a-bi is also a zero of multiplicity m of the polynomial P(x), and (x^2-2ax+a^2+b^2 )^m is a factor of P(x).

Müller’s method uses three initial approximations, p_0,p_1,and p_2 and determines the next approximation p_3 by considering the intersection of the x-axis with the parabola through(p_0,f(p_0 )), (p_1,f(p_1 )), and (p_2,f(p_2 )). (See Figure 1 (b).)