Math 351 Introduction to Numerical Analysis Summer 2009 Sample

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```					   Math 351                   Introduction to Numerical Analysis                 Summer 2009

Sample Problems for Final Exam

Recall that the ﬁnal exam is on Friday August 14th in our regular class room from 11:00 -
12:20. As for the midterm, the ﬁnal exam is open book and open notes. You should expect a bigger
emphasis on Numerical Integration and Numerical Linear Algebra. Please use list prepared for the
Midterm to review the topics that were covered there.

1. Using the data in Table 5.1 and Table 5.2, explain the diﬀerence in rate of convergence for
the Trapezoidal and Simpson’s Rule.

2. Recall that Simpson’s rule can be obtained applying Richardson extrapolation to the Trape-
zoidal method. In problem 10, page 202 what is known as ”Boole’s Rule” is introduced by
approximating the function f (x) with a polynomial of degree four. Show that the same for-
mula can be obtained by applying Richardson extrapolation to the Simpson’s rule. In other
words, show that the formula in the book can be obtained as

1
S4 (f ) +      (S4 (f ) − S2 (f )).
15
3. Do Problems 11, 12 and 13 in Section 5.1 of the text.

4. Do Problems 5 and 10 (c) in Section 5.2 of the text.

5. Do Problem 19 in Secton 5.2 of the text. Use Richardson extrapolation to obtain a better
estimate for the integral.

6. Do problems 1, 2 and 4 in Section 6.1 of the text.

7. Do problems 1, 2, 3, 4, 8, 21 and 30 in Section 6.2 of the text.

8. Do problems 4 and 5 of Section 6.3.

9. Do problems 3, 4 and 5 of Section 6.5.

10. Find the condition number of a diagonal matrix in terms of the elements in the diagonal.

11. Do problems 4, 10 an d 11 from Section 7.1 of the text.

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