Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

False-Position - Numerical Methods and Computing - Old Exam Paper, Exams of Mathematical Methods for Numerical Analysis and Optimization

Main points of this past exam are: False-Position, Bisection, Newton, Locating Single Roots, Roots of Equation, Newton’s Method Converges, Newton’s Method, Gauss Seidel Method, System of Linear Equations, Use of Over-Relaxation

Typology: Exams

2012/2013

Uploaded on 03/28/2013

shona
shona 🇮🇳

4

(1)

18 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
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 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)
pf3
pf4
pf5

Partial preview of the text

Download False-Position - Numerical Methods and Computing - Old Exam Paper and more Exams Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

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 fifi 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

hf^ 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

hf^ 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.