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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
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 Gauss-Seidel iterations start-
ing at x(0) = (O, 0)T. 8. (10 points)Let ...
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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