Cork Institute of Technology

Bachelor of Engineering in Structural Engineering-Stage 2

(CSTRU_8_Y2)

Summer 2009

Numerical Methods and Computing II

Legacy exam

(Time: 3 Hours)

Instructions Examiners: Dr. T. Creedon

Answer any four questions. Dr. P. Robinson

All questions carry equal marks.

Q1. (a) Describe any two of the following methods for obtaining roots of an equation:

(i) Bisection

(ii) False-Position

(iii) Newton (8 marks)

(b) Write a FORTRAN program for locating single roots using one of the

methods in part (a). (7 marks)

(c) Suppose 0)( =xf has a single root. Show that if )(xf and its derivatives are

continuous on an interval about the root and

()

1

)(

)()(

2

'

''

<

xf

xfxf for all x in this

interval, then Newton’s method converges to the root. (7 marks)

(d) Illustrate using a suitable example an equation with multiple roots. Describe

the modified Newton’s method for obtaining multiple roots. (3 marks)

Q2. (a) Describe the Gauss Seidel method for solving a system of linear equations.

(9 marks)

(b) Outline the general structure of a program for solving systems of linear

equations using the Gauss Seidel method. (8 marks)

(c) Describe the use of over-relaxation to improve the rate of convergence of the

Gauss Seidel method. (8 marks)

Q3. (a) Describe Lagrange interpolation referring to a general formula for )(xPn.

(6 marks)

(b) Given the data

Calculate (3.0)f using a Lagrange interpolating polynomial of degree 4.

(6 marks)

(c) Outline the general structure of a program for implementing Lagrange

interpolation. (6 marks)

(d) Given the data in the table below, approximate )5.2(f using a 3rd degree

Newton-Gregory interpolating polynomial. Estimate the error in your

approximation.

(7 marks)

x 1.0 2.7 3.2 4.8 6.4 8.0

)(xf 14.2 17.8 22.0 38.3 60.2 82.4

x 1.0 2.0 3.0 4.0 5.0

)(xf 10.1 20.3 43.1 52.2 61.2

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Numerical Methods

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28/03/2013