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dfswe - CMSC/MAPL 460 Dr Wolfe FINAL EXAM 1(10 points...

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Unformatted text preview: CMSC/MAPL 460 Dr. Wolfe FINAL EXAM July 27, 2001 1. (10 points) Suppose you have a computer with machine epsilon e = 10'”. Let ﬂm=mu+ﬁi (a) If x = 10"4 what result would the computer give for f (3:) ? (b) Find a way to compute f (104) to full machine accuracy and carry out the computation. __ 4 ~—2 _ 6 2. (15 points) Let (a) Solve Ax = b, using the LU factorization, forward elimination and back substitution. (b) Find an approximate solution to Ax = b by doing two Jacobi iterations starting at (0) = (3 2)T ' x , . (c) Find an approximate solution to Ax = b by doing two Gauss—Seidel iterations starting at x(0) = (3, 2)T. 3'. (15 points) Determine by two methods the polynomial of degree _<_ 2'wh0se graph passes through the points (1, 2), (2, 6) and (3,0). Verify that both methods give you the same answer. 2 d a: I=/~a 1 cc 4. (20 points) Let (a) Compute T4, the 4—panel trapezoid rule approximation to I. Compare your answer with the exact value of I. (b) Compute 0T4, the 4—panel corrected trapeziod rule approximation to I. Compare your answer with the exact value of I. ((2) Suppose you computed T16, the 16-panel trapezoid rule approximation to I. What answer would you expect (approximately) ? Do not actually carry out the computa— tion. ((1) How many panels would you need to compute I with an error of < 10‘6 using the trapezoid rule ? ' 5. (10 points) Let :63 + 6:1: 3x2 + 2' 9(x) = (a) Show that a _-= \/§ is a ﬁxed point of g. (b) Let 230 = 1.5. Compute \$1,332 and x3 for the ﬁxed point iterations xn+1 = g(xn). (c) Prove that if :00 if chosen sufﬁciently close to a, the ﬁxed point iterates converge to a (at least) quadratically. 6. (15 points) Consider the nonlinear system (a) Find all solutions to the system. (It might help to draw a picture.) (b) We Wish to ﬁnd a solution to the system by Newton’s method. If (xmyo) = (1,1) What is (.1.-1,311) ? Do not reduce to a single equation. 7. (15 points) Consider the initial value problem d _y = tyzi y(1) = 2' (a) Verify that the solution is Y(t) = 31%;. Find approximations to Y(1.2) by using (b) two steps of the Euler method with h = .1. (c) one step of the Improved Euler method with h z: .2. In (b) and (0) compare your answers with the exact solution. ...
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