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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering – Stage 2
(CSTRU_8_Y2)
Autumn 2008
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). (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.
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)
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) Outline the general structure of a program to implement Newton’s interpolating polynomial.
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
∫ 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.
(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.
Formulae:
Least Squares Correlation coefficient is
2 2 2 2
X Y XY r M X Y X Y M M
−
− −
Standard Error is 2
.. 1
S E e^ i M n
∑
Numerical Integration
Trapezoidal rule :
( ) 2 ( 0 2 1 2 2 2 1 )
b n n a
h ∫ f^ x dx^ ≈^ f^ +^ f^ +^ f^ +^ L +^ f^ −+ f
Error in Trapezoidal Rule is 2 ' '^ ( ) 12
b a h f ξ
− where ξ is some number between a and
b.
Simpson’s^13 Rule :
b n n n a
h ∫ f^ x dx^ ≈^ f^ +^ f^ +^ f^ +^ f^ +^ f^ +^ L+^ f^ −^ +^ f^ −+ f
Error in Simpson’s Rule is 4 (4)
b a
h f ξ
− where ξ is some number between a
and b.
