VIEWS: 7 PAGES: 8 CATEGORY: Technology POSTED ON: 3/11/2010 Public Domain
Numerical Analysis Candidacy Test Fall 2004 Name: ............................................................. In the following you must show your work. You should have worked out 7 problems when you are done: one from each of section 1, 2, 3, and 4, and three from section 4. 1 1 Interpolation Do Problem 1 or Problem 2. Problem 1 Let p(x) denote the polynomial of degree ≤ 4 with p(0) = 0, p(π/2) = 1, p(π) = 0, p(3π/2) = −1, and p(2π) = 0. Prove that we get an error bound π π (x + π) x + 2 x x− 2 (x − π) |p(x) − sin(x)| ≤ 5! for x ∈ [0, 2π]. Problem 2 Find the Newton form of the unique polynomial p(x) of degree ≤ 5 such that p(2) = 3 p (2) = 13 p(1) = 2 p (1) = 0 p(0) = 1 p (0) = 1. 2 2 Solving Nonlinear Equations Do Problem 3 or 4. Problem 3 You want to solve x2 − 2 = 0 on the interval [1, 2]. 1. Why does the bisection method apply on [1, 2]. 2. Find the approximate solution using two iterations of the bisection method on[1, 2]. 3. What upperbound for the error is given by the bisection method? Problem 4 You want to solve x2 − 2 = 0 on the interval [1, 2]. 1. Find the approximate solution using two iterations of Newton’s method starting at x = 2. 2. Explain what quadratic convergence means and give a result making pre- cise the statement “Newton’s method is quadratically convergent under good circumstances.” 3 3 Singular value decomposition Do either Problem 5 or Problem 6. Problem 5 a) Give a statement of the singular value decomposition for a square matrix. b) What is the singular value decomposition of 1 1 B := 5 2 1 1 1 + 1 1 2 −3 . 3 −1 Problem 6 a) Give a statement of the singular value decomposition for a square matrix. b) What are the singular values of 1 0 . 1 1 4 4 Norms and Eigenvalues Do Problem 7 or Problem 8. Problem 7 Let a b A := . c d v Regard A as the map from R2 to R2 which sends x := to v := w av + bw . Let the norm ||x||∞ on both of the R2 s be the ∞-norm, i.e., cv + dw ||x||∞ = max{|v|, |w|}. 1. Give the deﬁnition of the associated matrix norm ||A||∞ for A. 2. prove that ||A||∞ = max{|a| + |b|, |c| + |d|}. Problem 8 You have a complex matrix, a1,1 · · · a1,n A= . . . . . . . . . . an,1 · · · an,n 1. Explain what the Gerschgorin disks associated to A are and what their relation to eigenvalues of A are. 2. Use this relation to prove that if A is diagonally dominant, i.e., if |ai,i | > |ai,j | for all i, then A is invertible. j=i 5 5 Miscellaneous Do three of the following problems Problem 9 Consider a real valued 3 times diﬀerentiable function on an in- terval (0, 2). Using Taylor’s theorem we have f (1 + h) − f (1) h = f (1) + f (1) + O(h2 ) (1) h 2 for all suﬃciently small h. This gives the approximation f (1 + h) − f (1) f (1) = + O(h) h for all suﬃciently small h. Use Richardson extrapolation on Eq.1 to ﬁnd an O(h2 ) approximation to f (1). Problem 10 Perform the following computations in a) 3 digit rounding arithmetic; and compute b) the absolute error, and c) the relative error in each case: 1. (13.4 − 0.04) − 12.8; 2. (13.4 − 12.8) − 0.04. Put your answers in the table below. expression to 3 digit rounding answer absolute error relative error be evaluated (1 point each) (1 point each) (1 point each) (13.4 − 0.04) − 12.8 (13.4 − 12.8) − 0.04 6 Problem 11 Consider the following initial value problem: y = −y 2 ; 0 ≤ x ≤ 1; y(0) = 1. Use Taylor’s method of order two to approximate the solution at 1. with h = 0.5; and 2. with h = 0.25; 1 3. The actual solution of the above diﬀerential equation is y(x) = . 1+x Compute the absolute error of the approximate solution at x = 0.5 for the approximations with h = 0.5 and h = 0.25. Problem 12 Let S(x) be a free cubic spline deﬁned on [−1, 1] with nodes x0 = −1, x1 = 0, x2 = 1, and S(−1) = 0, S(0) = 1 and S(1) = 2. Assume that S(x) is given by the polynomial S0 (x) = a(x + 1) + b(x + 1)3 on [−1, 0] and by the polynomial S1 (x) = 1+cx+3dx2 −dx3 on [0, 1]. Find the relations that the real numbers a, b, c, d satisfy, and use them to ﬁnd a, b, c, d, i.e., ﬁnd the actual real numbers. [Recall the free splines satisfy the conditions that S0 (−1) = 0 = S1 (1).] 7 Problem 13 Prove: Neville’s Recursion Formula: Let x0 , . . . , xn be distinct real numbers in the interval, [a, b]. Let f be a real valued function on [a, b], and let P (x) denote the unique polynomial of degree ≤ n such that f (xi ) = P (xi ) for i = 0, . . . , n. For each c = 0, . . . , n, let Pb(x) denote the unique polynomial c of degree ≤ n − 1 such that Pb(xi ) = f (xi ) for 0 ≤ i ≤ n with i = c. Then c for two distinct points xj and xk in the set {x0 , . . . , xn }: (x − xj )Pb(x) − (x − xk )Pb (x) j k P (x) = xk − xj or equivalently: x − xj x − xk P (x) = Pb(x) + j Pb (x). xk − xj xj − xk k 8