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Math 128A - Fall 1997 - Rieffel - Final

Math 128A - Fall 1997 - Rieffel - Final - FRI 11:33 FAX...

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Unformatted text preview: 08/11/2000 FRI 11:33 FAX 6434330 MOFFITT LIBRARY M. Rieffel Mathematics 128A EnaJBcamjmflon December 18, 1997 SHOW YOUR WORK COMYANDNEATLY. Total points = 140. 1.Let p bethepolynomialwhichinterpolates f(x)=1/; atlhe 15 points 4, 9, and 16. firmlanupper boundfor |f(x)-p(x)l for 9 s x s 10. Justifyyour answer. 3 2' 2.121: A=[_l 4]. 15 a) Find the LU decomposition of A. . 9 b) Use the LU decomposition of A to solve the equation Ax = b 1 for b=[_1] . 1 5 c) Obtain an estimate for the condition number of A for a norm of your choice (specifirit). Justify your answer. 3. a) Describe briefly the strategy for deriving the Runge—Kutta '3 7 4 methods for solving ODE's. b) Explain precisely the derivation, following your strategy above, 14 of the second order RungeKutta meflmd c) Define what is meant the local truncation error for a 3 single-step method for solving ODE‘S. ' d) Show that the local truncation error for the above method is 12 of order n3 . . (over) 001 08/11/2000 FRI 11:33 FAX 6434330 MUFFITT LIBRARY 4. a) Give a brief but precise geometric explanation of how one 8 obtains the formula for the Newton-Raphson method for finding the zeros of a function. (Include a precise statement of the formula). h) Define precisely what it means for arconvergent sequence of 3 numbers to converge quadratically. c) Show precisely why the Newton-Raphson method often gives 1 2 quadratic convergence. - S. The Laguerre polynomials, L. , are orthogonal polynomials on 15 [0, +co) for the weight function w(1)= e" . They satisfy the beautiful recursion relation A (n+11L,+1(x) = (211-4-1— x).L.(x) - alrfix), while Lem-=1 and le-l-x. Derive the Gaussian two-point integration formula for g f(x)e"‘dx . 6. Suppose you are equipped with a pocket calculator with trig 1 5 functions (but without an integrate key), and you need to find fimxzmx withan errorof < 10". Brphinpredsdyhow you Would proceed to do this, eg. at what points you would evaluate 009(3) and what you would do with the values. (You do not need to carry out the computation.) Prove that your procedure will give an answer of the required accuracy. 002 ...
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