AMERICAN UNIVERSITY OF BEIRUT
Faculty of Arts and Sciences
Computer Science Department
CMPS 251
FINAL EXAM
FALL 2003-2004
Closed Book, Two hours
GIVE YOUR ANSWERS ON THE QUESTION SHEET
SUBMIT WITH BOOKLET
STUDENT NAME
ID NUMBER
1
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CMPS 251 Final Exam - Approximation and Numerical Integration, Exams of Numerical Methods in Engineering
The final exam questions for the Computer Science Department's CMPS 251 course at the American University of Beirut, Fall 2003-2004. The exam covers topics such as polynomial approximation, numerical differentiation using central difference formula and Richardson's extrapolation, and numerical integration using Mid-point, trapezoidal, Simpson's rules, and Romberg integrations.
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AMERICAN UNIVERSITY OF BEIRUT
Faculty of Arts and Sciences Computer Science Department
CMPS 251
FINAL EXAM
FALL 2003-
Closed Book, Two hours
GIVE YOUR ANSWERS ON THE QUESTION SHEET
SUBMIT WITH BOOKLET
STUDENT NAME
ID NUMBER
- In this problem, we consider the following table for the function f (x). All computations shall be carried out with 8 significant figures.
xi yi 0.000 4. 0.125 3. 0.250 3. 0.375 3. 0.500 3. 0.625 2. 0.750 2. 0.875 2. 1.000 2.
(a) Write the polynomial of degree 3 p(x) that would best approxi- mate f (0.3). Find p(0.3).
(c) Using the Mid-point, trapezoidal and Simpson’s rules followed by Romberg integrations fill out the following table used to approxi- mate I =
∫ (^1) 0 f^ (x)dx
h Mh Th Sh R(1) h R(2) h h 0 × × × × h 0 2 × × h 0 4 h 0 8 ×^ ×^ ×^ × × × × ×
- Suppose a real number L is appoximated by φ(h) such that:
L = φ(h) + c 1 h^3 + c 2 h^5 + c 3 h^7 + ...,
where the coefficients {ci} are independent from h. What combina- tion of φ(h) and φ(h 2 ) would give a better approximation φ^1 (h) to L than φ(h)? What is the order α of the approximation of L by φ^1 (h), (i.e. L = φ^1 (h) + O(hα))?
- To perform Naive Gauss elimination for the following quadridiagonal matrix (^)
d 1 u 1 v 1 0 .. .. 0 0 l 1 d 2 u 2 v 2 0 .. .. 0 0 l 2 d 3 u 3 v 3 0 .. 0 0 0 l 3 d 4 u 4 v 4 .. 0 :....... 0 .. 0 li di... :....... :... ln− 3 dn− 2 un− 2 vn− 2 0.... ln− 2 dn− 1 un− 1 0 .. .. 0 .. 0 ln− 1 dn
One uses the following algorithm: for i from 1 to n- li = li/di di+1 = di+1 − li ∗ ui if i<n- ui+1 = ui+1 − li ∗ vi end end Give the exact number of floating point operations needed to per- form this algorithm.
- Give with justification the minimum number of arithmetic operations (additions, subtractions and multiplications) to compute, using nested multiplication, the following polynomials
Polynomial p(x) Minimum number of arithmetic operations (x − 2)^17 + (x − 2)^31
4 x^5 − 6 x^12 + 2x^17 − x^33