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M.S. Numerical Exam 2008 (Department of Mathematical Sciences, M.S.U.) Instructions: Attempt all questions. Show all work. Carefully Read and Follow Directions. Clearly label your work and attach it to this sheet. 1. (a) Given the system of nonlinear equations below, give the general formulation of Newton’s Method for approximating a solution to this system. f1 (x1 , x2 , . . . , xn ) = 0 f2 (x1 , x2 , . . . , xn ) = 0 .. . fn (x1 , x2 , . . . , xn ) = 0 (b) Use Newton’s Method with x(0) = 0 to compute x(1) for the given nonlinear system. 1 4x21 − 20x1 + x22 + 8 = 0 4 1 x1 x22 + 2x1 − 5x2 + 8 = 0 2 2. Prove that if A ∈ IRn×n is nonsingular, then AAT is positive definite. 3. This problem deals with the vector norms denoted by k · kK , where K = 1, 2 and K = ∞. Let x ∈ IR2 , and define the unit ball induced by each of these norms in the following way, BK = {x ∈ IR2 : kxkK ≤ 1}, for K = 1, 2, ∞. Sketch each of these unit balls. 1 4. Suppose that one is constructing a quadrature rule to approximate the integral −1 f (x)dx. Find the values of w1 , w2 , w3 and w4 so that the following quadrature rule has degree of precision 3. Q(f ) = w1 f (−1) + w2 f (1) + w3 f 0 (−1) + w4 f 0 (1) R 5. Use Taylor Series expansions to derive the following order O(h4 ) formula for approximating f 0 (x). f 0 (x) = 1 [−f (x + 2h) + 8f (x + h) − 8f (x − h) + f (x − 2h)] + O(h4 ) 12h 6. Consider the IVP given below y 0 = f (t, y) y(t0 ) = y0 (a) Suppose a two-step method for numerically approximating the solution to the IVP is given by yk+1 = yk + ahf (tk , yk ) + bhf (tk−1 , yk−1 ), where a and b are constants. Give the expression for the Local Discretization Error at the k + 1st step. (b) Derive the Adams-Bashforth Two-Step Method by the following approach. Find a and b so that the method given in part (a) has Local Discretization Error of O(h3 ). 7. Define the term stiffness in the context of ODE IVPs. Which of the following methods, forward Euler’s method or backward Euler’s method, is more appropriate for stiff ODE IVPs? Explain why. Which method is more appropriate for ODE IVPs which are not stiff? Again explain why. 2