Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering – Stage 2
(Bachelor of Engineering in Structural Engineering – Stage 2)
(NFQ – Level 8)
Autumn 2005
Numerical Methods and Computing II
(Time: 3 Hours)
Instructions Examiners: Dr. T. Creedon
Answer any four questions. Prof. P. O’Donoghue
All questions carry equal marks. Mr. T. Corcoran
Q1. (a) Describe any two of the following methods for obtaining roots of an equation:
(i) Bisection
(ii) False-Position
(iii) Newton (7 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. (4 marks)
Q2. (a) Describe the Gaussian Elimination 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 either Gaussian elimination or the Gauss-Seidel method.
(8 marks)
(c) Describe the main pitfalls in using Gaussian elimination and list some
techniques for improving the solution. (8 marks)
