# Newton’s Method - Numerical Methods and Computing - Old Exam Paper, Exams for Mathematical Methods for Numerical Analysis and Optimization. Delhi Technological University

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Main points of this past exam are: Newton’s Method, Fortran Program, Rate of Convergence, System of Linear Equations, Lagrange Interpolation, Newton-Gregory Interpolating Polynomial, 3rd Degree Divided Difference, Centra...
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Microsoft Word - CSTRU_8_Y2 Numerical Methods & Computing II.DOC

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 )( 4hO 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

(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 ( ) cosf x x= between 0=x and x π= . (8 marks)