(Student Research Project) Muller’s method is an algorithm for computing solutions of an...

(Student Research Project) Muller’s method is an algorithm for computing solutions of an equation f (x) = 0. It is similar to the secant method in that it replaces f locally by a simple function and finds a root of it. Naturally, this step is repeated. The simple function chosen in Muller’s method is a quadratic polynomial, p, that interpolates f at the three most recent points. After p has been determined, its roots are computed, and one of them is chosen as the next point in the sequence. Since this quadratic function may have complex roots, the algorithm should be programmed with this in mind. Suppose that points x_n−2, x_n−1, and x_n have been computed. Set

where a, b, and c are determined so that p interpolates f at the three points mentioned previously. Then find the roots of p and take x_n+1 to be the root of p closest to x_n. At the beginning, three points must be furnished by the user. Program the method, allowing for complex numbers throughout. Test your program on the example

If the first three points are 1, 2, 3, then you should find that the polynomial is p(x) = 7(x−3)(x−2)+14(x−3)−4 and x₄ = 3.17971 086. Next, test your code on a polynomial having real coefficients but some complex roots.