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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2010
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
- Using Lagrange’s method for the following data
i 0 1 2 xi 1 2 3 f (xi) 2 1 4
(a) find the piecewise linear interpolating function; [5 marks] (b) find the quadratic interpolating polynomial. [5 marks]
- (a) Use the composite trapezoidal rule to estimate the integral ∫ (^1)
0
dx 1 + x with step sizes of h = 0.25 and h = 0.125. [6 marks]
(b) What is the error in each case? [4 marks] (c) Use one step of Romberg Integration to improve your answer. [3 marks]
- Write down the iteration scheme given by Newton’s method for determining a root x∗^ of the equation x^2 − 10 = 0. [3 marks]
Show that this is a second order method for determining x∗^ =
- [4 marks]
Starting with x 0 = 3, use this method to give an approximation to
10 that is correct to four decimal places. [5 marks]
(a) Show that the function f (x) = exp(−x) − x, has a root in the interval [0, 1]. [3 marks] (b) Describe the bisection method for determining a root of the equation f (x) = 0. [3 marks] (c) Use the bisection method to find this root to an accuracy of 2 decimal places. [8 marks]
Consider the following initial value problem:
y′^ = 1 + x sin(xy), 0 ≤ x ≤ 2 , y(0) = 0.
(a) Write down the difference equation given by Euler’s method with h = 0.5. [3 marks] (b) Use Euler’s method to estimate the value of the solution at x = 2. [4 marks] (c) Prove that the initial value problem has a unique solution for 0 ≤ x ≤ 2. [4 marks]
- Determine the values w 0 , w 1 and w 2 to obtain the integration rule ∫ (^) π
0
f (x) cos xdx = w 0 f (0) + w 1 f
(π
2
where E is zero for any quadratic polynomial f. [12 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 =
x^2 r +
xr
(b) xr+1 = 3 −
x^2 r
is suitable to compute a root of the equation
x^3 − 3 x^2 + 2 = 0
in the interval [2, 3]. [6 marks]
Using this method and a suitable starting point, compute the root to 3 decimal places. [5 marks]
- Consider the two-stage Runge-Kutta method
yn+1 = yn +
h 4
(k 1 + 3k 2 ) ,
where k 1 = f (xn, yn) ,
k 2 = f
xn +
h, yn +
hk 1
for solving the differential equation
y′^ = f (x, t).
(a) By considering its local truncation error, show that the method is at least of order 2. [8 marks]
(b) Investigate the stability of this method when used to solve y′^ = −λy(λ > 0). [6 marks]
