Math 128A - Fall 1998 - Rieffel - Final

# Math 128A - Fall 1998 - Rieffel - Final - FRI 10:27 FAX...

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Unformatted text preview: 08/11/2000 FRI 10:27 FAX 6434330 MDFFITT LIBRARY M. Rieffel Mathematics 128A FinalExamination December 17. 1998 snow YOUR WORK COWLETELYANDNEATLY. Total points= 140. 1. Give an example of an invertible 3x3 matrix A , a vector b , 18 and an approximate solution of AX 2: in whose residual error is 5 10"1 but whose error is e 1. Justify your answer. (You may use any norm as long as you specify it.) 2. a} Describe brieﬂy the strategy for deriving the Runge—Kutta 4 methods for solving ODE’s. b) Deﬁne what is meant by the local truncation error, and the 3 localorder for a single-step method for solving ODE's. c.) Derive a speciﬁc Runge-Kutta method of local order 3. Include 20 a precise explanation of how you know that your method is of local order 3. 3. a) Define precisely what it means for a c0nvergent sequence of 3 numbers to converge linearly. b) View gar) = ‘h‘i/(3X-i- I ) as an iteration function. Note that 1 15 is a ﬁred-point for g. Prove that for any initial guess which is is >- 1 the resulting iteration sequence will converge linearly to 1. ‘ (over) 001 08/11/2000 FRI 10:27 FAX 6434330 MDFFITT LIBRARY 1 3 4. a) Derive the simple Simpson rule for numerical integration. 4 by) Derive the composite Simpson rule. 3 c) Very brieﬂy discuss precisely the advantages of the composite Simpson rule compared to the composite trapezoid rule. 5. Find the LU decomposition of 17 A H H to H 43 IO N 1—} H Lu Check your answer. 6. Find a positive integer, n , such that if p is the polynomial 20 which interpolates f(x) = cos(x) at n equispaced points in the interval [3, 5] , then lﬂ’X) — p(X) I < 10"“ for every x in that interval Justify your answer. 20 7. Let p beapolynomial of degree n + 1 which, for the interval [2, 6] and the weight function w(x) = l, is orthogonal to all polynomials of lower degree. Assume that you have already proved that p then has n distinct roots in the interior of [2, 6]. Prove directly from this that the integration rule obtained I by interpolating functions at the roots of p and integrating over [2. 6] is exact for polynomials of degree 5 2n + l . 002 ...
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