Reference no: EM132503003

SEHS4612 Numerical Methods for Engineers Assignment - The Hong Kong Polytechnic University, Hong Kong

Instructions - Answer all questions in this paper. In Question 2(b), the matrix A is the 3-by-3 coefficient matrix in the linear system in Question 2(a). Show all your workings clearly and neatly. Reasonable steps should be shown.

Question1 -

(a) Find the third-order Taylor polynomial P₃(x)and the remainder term R₃(x)for the function

f(x) = xe^2x

expanded about a = 0. Determine an upper bound for R₃(0.2), and compare with the exact absolute error by considering the approximation of f(0.2) by P₃(0.2).

(b) Solve the following system of equations using Gaussian elimination with partial pivoting. All calculations should be kept to 4 significant figures.

x - 9y + 2z = 1

2x + 3y + 6z = 31

8x + 2y + 3z = 30

Question 2 -

(a) Solve the following linear system by the method of LU decomposition.

(b) Use result of (a) to find det(A).

(c) Consider the system Ax = b where

Starting with x₁ = 1, x₂ = 1, x₃ = 1, carry out three steps of the Gauss-Seidel iteration for solving the system.

Question3 -

The values given in the following divided difference table are exact using 5 decimal places. It is kown that f(x) = a_nxⁿ + a_n-1x^n-1 + · · · + a₁x + a₀, a_n ≠ 0 and n < 8.

(a) Find A, B, C, D, E, F, and G.

(b) Find an approximate value for f(0.65)by quadratic interpolation using Newton's Divided Difference method.

(c) Find an approximate value for f(0.22) by quadratic interpolation using Lagrange method.

(d) Determine n and a_n.

(e) By means of Newton's interpolating polynomial or otherwise, show that a_n is equal to the nth divided difference.

Question 4 -

The load W on a circular disc is given by the formula

W = sinθ/θ+1, 0 ≤ θ ≤ 2π, (*)

Where θ is the polar angle in radians. The graph of W is sketched in Figure 1. Suppose that the maximum load occurs when θ = α and the minimum load at θ = β.

(a) Show that both α and β satisfy the equation

tanθ = θ + 1

(b) Find α, accurate to 6 decimal places, by Newton's method on (*) using the starting value θ₀ = 1. What is the maximum load on the disc?

(c) Find β, accurate to 4 decimal places, by a fixed-point method other than Newton's method using the starting value θ₀ = 4. What is the minimum load on the disc?

(d) What is the definition of the order of convergence of an iterative method? What are the order of convergence of the methods used in part (b) and (c)? Give reasons.