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Cork Institute of Technology
Bachelor of Engineering in Structural Engineering-Stage 2
Numerical Methods and Computing II
(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 ( )
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.
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
Q4. (a) State the formula for Newton’s interpolating polynomial )(xPn of degree n. Derive this formula for the case 2=n . (8 marks) (b) Given the data in the table below, approximate (3)f using a 3rd degree
divided difference polynomial. Estimate the error in your approximation.
x ( )f x 3.2 22.0 2.7 17.8 1.0 14.2 4.8 38.3 5.6 51.7
(c) Outline the general structure of a program to implement Newton’s interpolating polynomial. (8 marks)
Q5. (a) Use central difference formulae to estimate the first and second derivative of 3( )f x x= at 5.1=x using a step size of 5.0=h . State the order of the error
for each estimate. (8 marks) (b) Use Richardson’s extrapolation to obtain an )( 4hO estimate of the first derivative of 3( )f x x= at 5.1=x . (8 marks) (c) Show that the differential operator D is related to the difference operator ∆ by:
)1ln(1 ∆+= h
Hence, show that the nth derivative can be approximated as follows:
ni n f
h fD ∆= 1 . (9 marks)
Q6 (a) Use the Trapezoidal rule and Romberg integration to find (correct to five decimal places)
xe dx−∫ . Start with 0.65h = .
(9 marks) (b) Apply the Trapezoidal rule and Simpson’s 1
3 rule to the data of the table below
to estimate 2.1
( )f x dx∫ .
i ix if if∆ 0 0.7 0.64835 0.26525 1 0.9 0.91360 0.24732 2 1.1 1.16092 0.20086 3 1.3 1.36178 0.13322 4 1.5 1.49500 0.05507 5 1.7 1.55007 -0.02125 6 1.9 1.52882 -0.08369 7 2.1 1.44513
(8 marks) (c) State the two-point Gaussian quadrature formula. Use two-point Gaussian quadrature to evaluate the integral of ( ) cosf x x= between 0=x and x π= . (8 marks)