Exmintion 1 Posted 23 October 2002. Due no lter thn 5pm on Mondy, 28 October 2002. Instructions: 1. Time limit: 3 uninterrupted hours. 2. There re four questions, plus bonus. Do not look t them until you begin the exm. 3. You my not use ny outside resources, such s books, notes, friends, clcultors, or MATLAB. 4. Plese nswer the questions thoroughly; show ll your work in detil to receive prtil credit. 5. Print your nme on the line below: 6. Indicte tht this is your own individul effort in complince with the instructions bove nd the honor system by writing out in full nd signing the trditionl pledge on the lines below. 7. Stple this pge to the front of your exm.
1. [20 points] The bisection, regul flsi, Newton, nd secnt methods re used to find the root of some prticulr function f C(R). Grphs showing the convergence behvior obtined for ech method re shown below, but the order is scrmbled. Bsed on typicl behvior of these lgorithms, mtch up ech plot with the pproprite method. Justify your nswers. A B f(x k ) f(x k ) 0 5 10 15 itertion, k 0 5 10 15 20 itertion, k C D f(x k ) f(x k ) 0 10 20 30 40 50 60 itertion, k 0 5 10 15 itertion, k 1
2. [25 points] This problem concerns the pproximtion of f(x) = x sin(x) on [ π/2, π/2]. () Write down, in the Lgrnge bsis, the qudrtic polynomil tht interpoltes f(x) = x sin(x) t the points x 0 = π/2, x 1 = 0, nd x 2 = π/2. (b) Repet prt (), but now using the Newton bsis. (c) Write down the stndrd formul for the interpoltion error f(x) p(x). (d) Apply this result to f(x) = x sin(x), nd from this formul, rgue tht the interpoltion error for f(x) = x sin(x) must converge to zero for ll x [ π/2, π/2] s the number of interpoltion points on [ π/2, π/2] grows. 2
3. [25 points] Consider the ordinry differentil eqution initil vlue problem x (t) = f(t, x), x(t 0 ) = x 0. The implicit Euler method is defined s x k+1 = x k + hf(t k+1, x k+1 ). () Assuming sufficient differentibility of x, write Tylor expnsion for x(t k ) expnded bout the point t k+1 = t k + h. (b) The trunction error for generl implicit one-step method x k+1 = x k + hφ(t k+1, x k+1 ; h) is given by T k = x(t k+1) x(t k ) h Φ(t k+1, x(t k+1 ); h). Using your result from (), clculte the trunction error for the implicit Euler method. 3
4. [30 points] For given weight function w(x) 0 on the intervl [, b], consider the inner product for ny f, g C[, b]. f, g := f(x) g(x) w(x) dx () Wht does it men for {φ j } n j=0 to be system of orthogonl polynomils? (b) Recll tht monic polynomil of degree n hs the form x n + c n 1 x n 1 + + c 1 x + c 0. Use the definition of orthogonlity to compute monic orthogonl polynomils of degree 0, 1, nd 2 for the weight function w(x) = 1 on the intervl [0, 1]. (c) Notice tht if {φ j } n j=0 is system of monic orthogonl polynomils, then ny monic polynomil q P n cn be written s n 1 q = φ n + γ j φ j for some constnts γ j, j = 0,..., n 1. (You do not need to prove this.) Using this expression for q, show tht j=0 or, equivlently, n 1 q, q = φ n, φ n + γ j φ j, γ j φ j, j=0 q(x) 2 w(x) dx = φ n (x) 2 w(x) dx + n 1 j=0 γ 2 j φ j (x) 2 w(x) dx. (d) The weighted L 2 norm is defined s ( f L 2 w := f, f 1/2 b 1/2. = f(x) 2 w(x) dx) From the result in prt (c), deduce tht φ n hs the smllest L 2 w norm of ny degree-n monic polynomil: φ n L 2 w = min q L 2 q P w. n q monic 4
5. [10 bonus points] This bonus question follows up on the result from question 4(d). The nottion is the sme s in tht question. Suppose we hve constructed polynomil p P n tht interpoltes some function f C n [, b] t the points {x j } n j=0. () The interpoltion error formul from question 2(c) leds to bound for the L norm of the interpoltion error, f p L 1 (n + 1)! mx x [,b] f n+1 (x) q L where q is some monic polynomil tht rises in the interpoltion error formul from question 2(c). Develop similr bound for the L 2 w norm of the error. (b) Use the result from question 4(d) to determine interpoltion points {x j } n j=0 tht optimize the L 2 w error bound developed in prt (). (c) How is this result relted to Gussin qudrture? 5