AMS 527 Sample Test 1 February 19, 2013 Note: The exams are closed-book. A one-page cheat sheet is allowed, but you must prepare the it yourself. 1. Answer whether the following statements are true or false and give a brief explanation. (a) The choice of algorithm for solving a problem has no effect on the propagated data error. (b) For a given fixed level of accuracy, a superlinearly convergent iterative method always requires fewer iterations than a linearly convergent method to find a solution to the level of accuracy. (c) If a function is unimodal on a closed interval, then it has exactly one minimum on the interval. (d) When using Newton’s method for solving a nonlinear equation, it may converge linearly, may converge quadratically, and may not converge at all. (e) For minimizing a real-valued function of several variables, the BFGS secant update method is always faster than Newton’s method. 2. For the approximation of the zeros of the function f (x) = (2x2 − 3x − 2)/(x − 1), consider the following fixed-point methods: (a) x(k+1) = g(x(k) ), where g(x) = (3x2 − 4x − 2)/(x − 1); (b) x(k+1) = h(x(k) ), where h(x) = x − 2 + x/(x − 1). Analyze the convergence properties of the two methods and determine in particular their orders of convergence near the solutions α1 = −1/2 and α2 = 2. 3. Newton’s method for solving a scalar nonlinear equation f (x) = 0 requires computation of the derivative of f at each iteration. Suppose that we instead replace the true derivative with a constant d, that is, we use the iteration scheme xk+1 = xk − f (xk )/d. (a) Under what condition on the value of d will this scheme be locally convergent? (b) What will be the convergence rate, in general? (c) Is there any value for d that would still yield quadratic convergence? 4. Consider the function f : R2 → R defined by f (x) = 1 2 1 (x1 − x2 )2 + (1 − x1 )2 . 2 2 (a) Use the first- and second-order optimality conditions to show that x∗ = [1, 1] is a local minimum. (b) Give the linear system for the first iteration of Newton’s method for minimizing f using x0 = [2, 2]T as starting point. (You do not need to solve the linear system by hand.) (c) Outline the first iteration of steepest descent method for minimizing f using x0 = [2, 2]T as starting point. (You do not need to solve the line-search step by hand.) 1