This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: AMSC/CMSC 460 Dr. Wolfe FINAL EXAM May 17, 2004 1.(10 points) Let f($) = x2 — 6.12: + 1.5. Then f(4.71) = —5.0469. Do the following in
three digit arithmetic with rounding: (a) Compute (4.71)2 ~ (6.1)(4.71) + 1.5. Compute the relative error in the result.
(b) Evaluate f (4.71) using Homer’s method (synthetic division). Compute the relative
error. What do you conclude? 2.(1O points)
(a) Let a: = 3 1020, y = 4 102°. Compute z r: V5132 + .112.
(b) In IEEE single precision ﬂoating point arithmetic the largest machine number is about 2127 z 1.70 1038. What would happen if we tried to compute 2 according to the
formula above in this arithmetic ? (c) Can you rewrite the formula for z in such a way that we can compute it in the above
arithmetic? 3. (15 points) Let 1 2 —l
A: 2 4 0
0 1 —1 (a) Show that we cannot write A = LU where U is upper triangular and L is lower
triangular with ones on the diagonal. ' (b) Find a permutation matrix P, upper triangular matrix U and a lower triangular matrix
L such that PA = LU.
(0) Explain how the result of part (b) is used to solve Ax = b. 4. (15 points) Given the data points (0,2), (%,5), (1,4).
(a) Find p2(:t), the polynomial of degee S 2 interpolating this data.
(b) Find the function P(a:) of the form P(:v) = A + Bcosn'w + C'sinvra: interpolating the data. 5. (12 points) Find constants A and B such that the integration rule h
[0 mmdx z hwm) + BMW is exact for all ﬁrst degree polynomials. 6. (13 points) (a) What are the solutions a, if any, of the equation x = 1 + :13 ? (b) Does the iteration can“ : x/ 1 + 113,, converge to any of these solutions (assuming 3:0
is chosen sufﬁciently close to a)? 7. (15 points)
(a) Transform the second order system of differential equations U” + 311’ — 2n = sint
v"+u’—u+v=cost (1)
u(0) == 1, u’(0) = 2, 21(0) = 3, v'(0) = 4 into a ﬁrst order system. (Here ’ = 5;).
(b) Compute an approximation to the solution of (1) at t = 0.1 using Euler’s method with h: .1.
6 —2 14 As you can easily check, the solution of Ax = b is x = (3, 2)T. (a) Find an approximate solution of Ax = b by doing three Jacobi iterations starting at
x(0) = (0, 0)T. (b) Find an approximate solution of Ax = b by doing three GaussSeidel iterations start
ing at x(0) = (O, 0)T. 8. (10 points)Let ...
View
Full
Document
This note was uploaded on 04/04/2010 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Algebra

Click to edit the document details