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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering- Stage 2
(NFQ Level 8)
Summer 2007
Numerical Methods and Computing II
(Time: 3 Hours)
Instructions Examiners: Dr. T. Creedon Answer any four questions. Mr. P. Anthony All questions carry equal marks. Prof. P. O’Donoghue
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). (8 marks)
(c) Suppose that (^) f ( x )= 0 has a single root. Show that if (^) f ( x )and its derivatives
are continuous on an interval about the root and
1 ()
( ) () '^2
'' < f x
f x f x for all x in
this interval and the initial value x (^) 0 is chosen in this interval, then Newton’s method converges to the root. (9 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. (7 marks)
(d) Describe cubic spline interpolation. (6 marks)
Q4. (a) State the formula for Newton’s interpolating polynomial Pn ( x )of degree n.
Derive this formula for the case n = 2. (8 marks)
(b) The points (1,0), (2, 0.693), (5,1.609) lie on the curve f ( ) x = ln x. Fit a 2nd
order divided-difference interpolating polynomial to the data and hence
calculate f ( 3 )=ln 3. Use the additional point (6,1.792) to estimate the error
in your calculation of f ( 3 ).
(9 marks)
(c) Given the data in the table below, approximate f ( 2. 5 )using a 3 rd^ degree
Newton-Gregory interpolating polynomial. Estimate the error in your
approximation.
(8 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 to estimate
1.5 2
e −^ xdx
∫ with
(i) h =0. (ii) h =0. State the order of the error in each case. (6 marks)
(b) Use Romberg integration and the estimates from part (a) to get an improved estimate of 1.5 2
e −^ xdx
What is the order of the error for this improved estimate? (6 marks)
(c) Apply the Trapezoidal rule and Simpson’s 1 3
rule to the data of the table below
to estimate
∫ f^ ( ) x dx.
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.
(7 marks)
(d) State the two-point Gaussian quadrature formula. Use two-point Gaussian
quadrature to evaluate the integral of f ( x )= sin x between x = 0 and 2
π x =.
(6 marks)
