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

Numerical Methods

Description: 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, Central Difference Formulae
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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
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
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)
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