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

xx= 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

xx= 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

n

i

nf

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

x

edx

−

∫.

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

0.7

()

f

xdx

∫.

i i

x

i

f

i

f

∆

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

xx

=

between 0

=

x and

x

π

=.

(8 marks)

##### Document information

Uploaded by:
shona

Views: 2776

Downloads :
0

University:
Delhi Technological University

Subject:
Numerical Methods

Upload date:
28/03/2013