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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2011
MA25110 - Introduction to Numerical Analysis
Time allowed - 2 hours
- All questions may be attempted.
- Marks gained from questions in section B will be given greater consideration in assessing a first class performance.
- Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
Section A
- Construct the forward difference table for the following data:
x 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 f (x) 0. 55 0. 82 1. 15 1. 54 1. 99 2. 50 What is the lowest degree of polynomial which matches the data exactly? [6 marks] Determine the interpolating polynomial using the forward difference formula
pn(x) =
∑^ n k=
(s k
∆kf (x 0 ),
where x = x 0 + sh. [5 marks]
(a) Use the composite trapezoidal rule to estimate the integral ∫ π 4 0 sin(2x)dx with step sizes of h = π 8 and h = 16 π. [6 marks] (b) What is the error in each case? [3 marks] (c) Use one step of Romberg Integration to improve your answer. [3 marks]
Simpsons rule for approximating the integral ∫^ ab f (x)dx is given by
S 2 f = h 3 (f (x 0 ) + 4f (x 1 ) + f (x 2 )) , where x 0 = a, x 1 = a+ 2 b, x 2 = b and h = b− 2 a. (a) By dividing the interval [a, b] into 2N subintervals of equal length h = b 2 −Na , derive the composite Simpsons rule S 2 N. [4 marks] (b) Use this rule to approximate the value of the integral ∫ (^1) 0 e^2 xdx with N =1 and N =2. [5 marks] (c) The error formula for the composite Simpsons rule is given by
ES 2 N f :=
∫ (^) b a f (x)dx − S 2 N f = − h
4 180 (b^ −^ a)f^
(4) (^) (η)
for some η ∈ (a, b). Give upper bounds on the error for your calculations in (b). [3 marks]
Section B
(a) Given the data: i xi yi 0 0 0 1 1 1 2 2 0 (i) Describe the properties of its cubic spline interpolant s(x). [4 marks] (ii) State the conditions that s(x) satifies if it’s a natural cubic spline. [2 marks] (b) Determine the natural cubic spline s(x) that interpolates the data. [12 marks]
Determine the values w 0 , w 1 and w 2 to obtain the integration rule ∫ (^2) 0 f (x)exdx = w 0 f (0) + w 1 f (1) + w 2 f (2) + E, where E is zero for any quadratic polynomial f. [14 marks]
State sufficient conditions for the sequence xr defined by
xr+1 = g (xr) , r = 0, 1 , 2 , ... with x 0 ∈ [a, b] to converge to a unique root x∗^ ∈ [a, b], of the equation x = g(x). [3 marks] Determine which of the two iterative methods (a) xr+1 =^12 (x^2 r − 9 ) (b) xr+1 = √ 2 xr + 9 is suitable to compute a root of the equation x^2 − 2 x − 9 = 0 in the interval [4, 5]. [6 marks] Using this method and a suitable starting point, compute the root to 3 decimal places. [5 marks]
- (a) Use the Euler-Trapezoidal predictor-corrector pair
y[ np+1] = yn + hf (xn, yn) yn+1 = yn + h 2
f (xn, yn) + f
xn+1, y n[p+1]
to solve the initial value problem y′^ = −y^2 , 1 ≤ x ≤ 2 , y(1) = 1, with h = 0. 25 [8 marks] (b) What restriction on h, if any, is required for this predictor-corrector method to be absolutely stable? [6 marks]
