Newton’s Method - Numerical Methods and Computing - Old Exam Paper, Exams for Numerical Methods. Delhi Technological University

Numerical Methods

Description: Main points of this past exam are: Newton’s Method, Fortran Program, Rate of Convergence, System of Linear Equations, Lagrange Interpolation, Newton-Gregory Interpolating Polynomial, 3rd Degree Divided Difference, Central Difference Formulae
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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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