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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

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

(9 marks)

(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 )( 4*hO * 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
*

*D *.

Hence, show that the nth derivative can be approximated as follows:

*i
n
*

*ni
n f
*

*h
fD *∆= 1 . (9 marks)

Q6 (a) Use the Trapezoidal rule and Romberg integration to find (correct to five decimal places)

2 1.5

0.2

*xe dx*−∫ .
Start with 0.65*h *= .

(9 marks) (b) Apply the Trapezoidal rule and Simpson’s 1

3 rule to the data of the table below

to estimate 2.1

0.7

( )*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 ( ) cos*f x x*= between 0=*x * and *x *π= .
(8 marks)