AMS 527 Sample Test 1 February 19, 2013

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.)
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