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Unformatted text preview: 09/27/2000 WED 11:06 FAX 6434330 MOFFITT LIBRARY 001 Math 128B: 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
ﬂoatingpoint 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 offdiagonal 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 sixteendigit 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 rankone 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 onedimensional 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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