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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
Typology: Exams
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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 f ( x )= 0 has a single root. Show that if f ( x )and its derivatives are
continuous on an interval about the root and
'^2
'' < f x
f x f x 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 Pn ( x ).
(6 marks) (b) Given the data
Calculate f (3.0)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 f ( 2. 5 )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. f ( x ) 14.2 17.8 22.0 38.3 60.2 82.
x 1.0 2.0 3.0 4.0 5. f ( x ) 10.1 20.3 43.1 52.2 61.
Q6 (a) Use the Trapezoidal rule and Romberg integration to find (correct to five decimal places) 1.5 2
e −^ xdx
Start with h = 0.65. (9 marks) (b) Apply the Trapezoidal rule and Simpson’s 1 3
rule to the data of the table below
to estimate
i (^) xi fi ∆ fi 0 0.7 0.64835 0. 1 0.9 0.91360 0. 2 1.1 1.16092 0. 3 1.3 1.36178 0. 4 1.5 1.49500 0. 5 1.7 1.55007 -0. 6 1.9 1.52882 -0. 7 2.1 1.
(8 marks)
(c) State the two-point Gaussian quadrature formula. Use two-point Gaussian quadrature to evaluate the integral of f ( ) x = cos x between x = 0 and x = π. (8 marks)
