Q1. Attempt all questions. The marks for each question are indicated against it.

• (a)) Explain round-off error and truncation error, with one example of each. (320 words)
• (b)) The series for e^x can be written as: e^x = 1 + x + x^2/2! + x^3/3! + ... Calculate the truncation error when the first four terms of the series are used to evaluate e^1.5 (320 words)
• (c)) Explain why a polynomial is a useful choice for an interpolating function. (160 words)

Q2. Use the Secant method with initial guesses of x0=1 and x1=2 to calculate the root/s of the equation f(x) = x^3 – x – 1 up to three iterations. Calculate the absolute relative approximate error at each stage. What are the advantages and drawbacks of the Secant method when compared to the Bisection Method and the Newton-Raphson Method?

• (a)) Use the Secant method with initial guesses of x0=1 and x1=2 to calculate the root/s of the equation f(x) = x^3 – x – 1 up to three iterations. Calculate the absolute relative approximate error at each stage. (480 words)
• (b)) What are the advantages and drawbacks of the Secant method when compared to the Bisection Method and the Newton-Raphson Method? (320 words)

Q3. A population of single-celled organisms was grown in a Petri dish over a period of 24 hours. The number of organisms at a given time is recorded in the table below. Fit the data in the table below to an exponential model y = a exp(x): x (time in hours) 0, 4, 8, 12, 16, 20, 24 y 25, 36, 52, 68, 85, 104, 142. Calculate the first derivative of the function f(x) = sin(2x) at x = π/3 using the forward divided difference, backward divided difference and central divided difference methods with a step size of h = 0.1 rad. Also calculate the absolute true error in each case.

Q4. Evaluate the integral from 1 to 2 of x^3 ln x dx using the 2-point Gaussian Quadrature Rule with the following values: Points 1, 2; Weights (Ci) 1.0, 1.0; Arguments (Xi) 0.577, -0.577. Derive the Runge-Kutta 2nd Order formula for the differential equation: dy/dx + y = exp(-x). Use this to determine the value of y at x = 2 given that y(x = 0) = 2, using a step size h = 1.0.

Q5. Using the linear congruential random number generator determine the first five random numbers given that: x0 = 27, a = 17, c = 43, and m = 100. Transform these random numbers to lie between 0 and 1. What would be the maximum possible period of this random number sequence? Use the LU decomposition method to solve the following system of equations: 2X1 + X2 - X3 = 2; X1 + 2X2 + X3 = 8; -X1 + X2 - X3 = -5.