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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2008
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
- Given the values of a function f on a set of n + 1 distinct points xi, i = 0, ..., n, describe Lagrange’s formula for the polynomial pn of order n that interpolates the data. [4 marks] Write out the formulae for the linear and quadratic interpolants p 1 and p 2 .[2 marks] For the following data i 0 1 2 xi 0 1 3 f (xi) 2 4 6
(a) find the piecewise linear interpolating function; [3 marks]
(b) find the quadratic interpolating polynomial. [3 marks]
- (a) Construct the divided difference table for the following data:
x 0. 0 0. 2 0. 3 0. 4 0. 7 1. 0 f (x) 1. 00 0. 8432 0. 8062 0. 8112 1. 2702 3. 00
What is the lowest degree of polynomial which matches the data exactly? [7 marks] (b) Determine the interpolating polynomial using Newton’s formula:
pn(x) = f [x 0 ] +
∑^ n
k=
f [x 0 , ..., xk]
k∏− 1
j=
(x − xj ).
[5 marks]
- Divide the interval [a, b] into N subintervals of equal length h.
(a) Derive the composite Trapezoidal rule TN for approximating the integral ∫ (^) b
a
f (x) dx.
[6 marks] (b) Use this rule to approximate the value of the integral ∫ (^) π/ 4
0
sin(2x) dx
with N = 2 and N = 4. [6 marks]
Section B
- Determine the values w 0 , w 1 and w 2 to obtain the integration rule ∫ (^2)
0
ex^ f (x) dx = w 0 f (0) + w 1 f (1) + w 2 f (2) + E,
where E is zero for any quadratic polynomial f. [12 marks]
- (a) 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] (b) (i) Determine which of the two iterative methods
(I) xr+1 = x
(^2) r − 7 2 (II) xr+1 =
2 xr + 7
is suitable to compute a root of the equation
x^2 − 2 x − 7 = 0
in the interval [3, 4]. [6 marks] (ii) Using this method and a suitable starting point, compute the root to 3 decimal places. [6 marks]
- (a) Write down the difference equation given by Euler’s method to approximate the solution of the initial value problem
y′^ = 2y + 1, 0 ≤ x ≤ 0. 5 , y(0) = 0. 25.
[3 marks] (b) Solve the difference equation to show that yn, the numerical approximation to y(x) at x = nh, is given by
yn = (1 + 2h)n
y 0 +
[8 marks] (c) Use the formula to obtain estimates to y(0.5) with the step sizes h = 0.1 and h = 0.05. [2 marks]
- (a) Given the data (0, 0), (1, 1), (2, 0), write down the conditions the natural cubic spline interpolant s has to satisfy. [8 marks] (b) Determine s. [10 marks] (c) Given that the polynomial p(x) = x(2 − x) is of degree no greater than 3 and fits the data, why isn’t s(x) = p(x)? [2 marks]
