Math 128B - Spring 1999 - Strain - Final

Math 128B - Spring 1999 - Strain - Final - 09/27/2000 WED...

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Unformatted text preview: 09/27/2000 WED 11:06 FAX 6434330 MOFFITT LIBRARY 001 Math 128-B: Spring 1999, J. Strain. Final Exam, 19 May 1999, 1230—1530. The following problems are worth 30 points each. Please solve enough to get 90 points. 1. Assume a fundamental matrix Y(t) for y’ = P(t)y is known on the interval [(1, b]. (a) Find a formula for the solution of the initial value problem y' 2 any + M) satisfying 31(0) 2 n. (b) Derive a formula for the Green function G (t, s) of the boundary value problem y'2P(t)y+f(t) a<t<b A3101) + 331(5) = 0, assuming A and B are n ><n matrices such that the matrix D = AY(o) +BY(b) is invertible. 2. Consider the nonlinear two‘point boundary value problem u”——exp(u)=0 0<m<1 11(0) 2 u(1) = 0. Discretize by centered differences with h = 1/ (N + 1) to get 51(0),; = vi+1 —- 211% + vi_1 ~— in2 expfiii) : 0 1’0 =9N+1 =0 Where 113- z (a) Write down Newton’s method for solving the nonlinear system F(v) : O. (b) Let B be the matrix of the linear system in (a). Prove that B is invertible. (c) Define and evaluate the local truncation error of the centered scheme above. ((1) Define and evaluate the “stability” of this scheme and explain how it relates to the accuracy of the numerical solution in the limit it —> 0. 3. Consider the linear ODE y”=q(=v)y 0<m<1 with 11(3) > 0 on 0 5 ac f 1, and the finite difi'crcnce scheme (qu — zuj + uj11)/h2 = qjuj 0 < j < N + 1 Where qj : q(jh) and (N+ 1)h = 1. 09/27/2000 WED 11:07 FAX 6434330 MOFFITT LIBRARY 002 (a) Let y be a solution of the ODE with y(0) = y(1) = 0. Prove or disprove: 3; must be identically zero. (b) Let u be a solution of the finite difference scheme with ug : uN+1 : 0. Prove or disprove: u must be identically zero. (c) Prove or disprove: the matrix —2 — fiqu 1 0 U 1 —2 — 15qu 1 0 S = 0 1 "—2 "- h2q3 1 D 1 —2 — liqu is invertible. 4. Let n m = 13:1 Decide which formula for the variance is likely to give a better relative error bound in floating-point arithmetic and explain the analysis behind your choice: (a) n (n — 1).S'2 = — nm2 i=1 or (b) ’ Tl. (n —u 1)S2 : — m)2. imi 5. Suppose A is an n by 71 real invertible matrix and 'r = h — Ay where b = A3: is nonzero. Prove that H39 - yll/llrfill S 00nd(A)||TH/|lbll-. 6. Prove that floating point arithmetic with machine epsilon 6 produces an inner product (summed in the natural order) Satisfying fl(xTy) 2 wily + C) where leii S as long as 116 g 1/2. Under what conditions on m and y does this bound guarantee small relative error in mTy? 09/27/2000 WED 11:07 FAX 6434330 MOFFITT LIBRARY 003 .7. Let 1 0 I A: 0 2 0 1 0 3 (a) Find a symmetric tridiagonal matrix T with the same eigenvalues as A. (b) Find an orthogonal matrix Q and an upper triangular matrix R with positive diagonal entries such that T —- T33I = QR. (c) Compute RQ + T33I and verify that it is closer to diagonal than T by computing the sum of squares of off-diagonal elements for both. 8. Let A be a number in the range 0.25 g A S 1 and consider the Newton iteration $k+1 = (me + Amid/2 for computing arm = x/A. Assume that the initial guess $0 = (1 + 2A) / 3 has error less than 0.05 for any such A. (a) Prove that the error 6;, = as}, — x/A satisfies 3k+1 = tag/217k- (b) Determine the number k of steps necessary to guarantee sixteen-digit accuracy in VA. 9. (a) Let A be an unknown matrix. Given a known matrix B and known vectors p and q = Ap, find a rank-one update C = B + uvT of B which makes IIC- Allz S HB'"AH2 and Up = q. Show that your update satisfies these two requirements. (b) Suppose the QR factorization of a nonsingular matrix A = QR is given, where Q is orthogonal and R is upper triangular with positive diagonal elements. Under what conditions on A, n and 'u will the rank—one update B : A + uvT have a QR factorization and What algorithmic sequence of steps can be used to find it? 10. For an arbitrary diiferentiable function F : R” —> R", write Newton’s method as a fixed point iteration xk+1 Z G Determine G in terms of F and DF. Now restrict your analysis to the one-dimensional case n = 1, assume F has two continuous derivatives, and use Taylor expansion to prove superlinear convergence: as any two distinct points a and 11 approach a zero of F where F’ is nonzero, |G(u) —— — v] —> 0. 11. Suppose a square matrix A has a factorization A m U EJVT where U and V are orthogonal and E is diagonal with diagonal elements 0'1 2 0'2 2 ...0',. > 0 = 0 = = 0. Use this factorization to find a vector at which minimizes subject to the requirement that HA3: — be minimum. ...
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