MATH 448-548 Midterm project assignment 1. Use Taylor formula to derive the error term for the approximate formula f ′(x) ≈ 1 2h (−3f(x) + 4f(x + h)− f(x + 2h)) . 2. Consider the following variation of the Newton’s method: xn+1 = xn − f(xn) f ′(x0) . Find constants C and s such that en+1 = Ce s n. 3. Consider the linear system Ax = b, where A = 6.25 −1 0.5−1 5 2.12 0.5 2.12 3.6 , and b = 7.5−8.68 −0.24 . Write a MATLAB program for LU-factorization with a unit lower triangular L (meaning that the diagonal entries should be equal to one). Then write a program for the Cholesky factorization. WARNING: avoid using MATLAB shortcuts. The programming should be done ”from scratch”. 4. Do Problem 5 from Computer Problems 2.2 (p. 62) in the book. You can use the previous Problem 4 for guidance. 5. Implement and analyze the Ridders’ method for solving the equation f(x) = 0. For guidance, you might wish to read the Wikipedia page on the Ridders’ method. Also feel free to consult other sources. Brief Description of the method. Given the initial bracketing interval [x0, x2], contain- ing one solution, and such that f(x0) and f(x2) have opposite signs, compute the midpoint x1 = (x0 + x2)/2. Then define a new function h(x) = f(x)e ax. Find the parameter a that ensures h(x1) = 1 2 (h(x0) + h(x2)). Then define x3 as the x-intercept of the line passing through (x0, h(x0)) and (x2, h(x2)). Use x3 as one of the endpoints of the next interval bracketing the solution. The other endpoint is x1 if f(x1)f(x3) < 0. Otherwise, choose either x0 or x2 based on the requirement that the sign of f(x) at the chosen point must be opposite to the sign of f(x3). Continue until the desired accuracy is reached. Questions and items to implement 1. Show that a is uniquely defined and give the equation for finding it. 2. Work out a formula for x3 in detail. 3. Implement Ridders’ method in MATLAB together with the standard bisection method for finding all solutions of ex − x2 = 0. 4. Numerically, compare the convergence rate of both algorithms. Discuss the results. What is the approximate order of convergence for the Ridders’ algorithm?
