MATH 251 Exam 2, Fall 2010
American University of Beirut · Numerical Computing · 12 pages
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AMERICAN UNIVERSITY OF BEIRUT
Mathematics Department-FAS
MATH 251
FALL 2010-2011
Quiz 2
Closed Book, 75 mn
STUDENT NAME
ID NUMBER
Problem
Out of
Grade
1
15
2
20
4
15
TOTAL
50
1
1. (15 points) Let A be the following 4 × 4 matrix:
2
1
−1
1
1
1
0
3
−1
2
3
−1
3
−1
−1
2
(a) (8 points) Apply on this matrix, Gauss Elimination with the
scaled partial pivoting strategy, showing the status of the
matrix after each elimination, i.e. each of the pivot rows and the
corresponding multipliers should be identified and circled. Update
the index vector at each reduction.
2
(b) (3 points) Extract the 4 by 4 matrices P, L and U which determine
the LU-factorization of A.
• P=
• L=
• U =
3
(c) (4 points) Use the LU - decomposition of A to determine the
last column of A−1.
4
2. (20 points) Consider the following set of data
D3 = {(0, 1); (1, 21); (2, 22); (3, 23)}
that corresponds to the function f(x) = 2x.
(a) (10 points) Based on the set D3, determine the equations of the
Natural Cubic spline function S(x), that approximate the function
f(x) = 2x.
5
....
6
(b) (5 points) Write the equations of S′(x) that approximate the
derivative f
′(x) on the interval [0, 3].
7
(c) (5 points)Use the Cubic spline S(x) and its derivative to approx-
imate f(0.5) and f ′(0.5) in F(10, 5, −20, +20) rounding to the
closest if needed. Calculate then the relative errors for both ap-
proximations.
• f(0.5) ≈
• Relative Error:
8
• f ′(0.5) ≈
• Relative Error:
9
3. (15 points) Consider the following set of data:
i
xi
f(xi)
0
0.000
1.000
1
0.125
1.110
2
0.250
1.197
3
0.375
1.266
4
0.500
1.319
(a) (5 points) Write first the Forward difference formula φh(f(xi))
that approximates f
′(xi) , then derive the expression of the error
series ϵ(h) = f
′(xi) − φh(f(xi)) in the form:
ϵ(h) = c1hα1 + c2hα2 + c3hα3 + ....,
by determining the values of the constants {α1, α2, α3, ...}.
• φh(f(xi)) =
• ϵ(h) =
10
(b) (5 points) Based on the Forward difference formula, derive Richard-
son extrapolation operators of orders 1 and 2 and the order of their
error series.
• φ1
h(f(xi)) =
Corresponding Error = O(...........)
• φ2
h(f(xi)) =
Corresponding Error = O(...........)
11
(c) (5 points) For the purpose of improving the approximation to
f ′(0.000), fill in the empty slots in the following table adequately,
starting with h0 = 0.5.
Express all the results obtained in F(10, 5, −15, +15).
h
φh(.)
φ1
h(.)
φ2
h(.)
h0 = 0.5
h0/2
h0/4
Best approximation to f ′(0.000) :
12