NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
B.Tech., M.Sc.(Integrated) END-SEMESTER (AUTUMN) EXAMINATION, 2019
Course ID.: MA-2305
Subject: Numerical Analysis
Duration of Examination: 3 Hours
Full Marks-50
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Answer ALL Questions
Each question carries equal marks
No part of a question may be answered separately
(Notations have their usual meanings)
1.
Find a root of the following equation 2 x − log10 x = 7 by Regula-falsi method taking initial guess x0 = 3
correct up to 3 significant digits.
2.
Find f (1955) using Newton’s backward method from the following table
x
f (x)
3.
4.
5.
1921
46
1931
66
1941
81
1951
93
1961
101
Solve the following system of equations by Crout’s method
2x − 2 y + z = 2
5x + y − 3z = 0 .
3x + 4 y + z = 9
-1
Find A by Gauss-Jordan method for the following matrix
2 4 5
A = 1 − 1 2 .
3 4 5
Find the largest eigenvalue (in magnitude) and corresponding eigenvector of the following matrix by Power
method
 1 3 2
A = − 1 0 2 ,
 3 4 5
performing first three iterations only, taking initial guess, x0 = {1 1 1}T .
1
6.
Compute the integral
∫
dx
by 3-point Gauss-Legendre quadrature formula.
1+ x4
Solve the following linear system of equations using Gauss-Seidel method
x1 + 9 x 2 − 2 x3 = 36
0
7.
2 x1 − x 2 + 8x3 = 121
6 x1 + x2 + x3 = 107 ,
starting with x (0 ) = [1, 1, 1]T , iterate three times.
P.T.O.
8.
Using Newton’s method for nonlinear systems, solve the following nonlinear system in two iterations
x2 + y2 = 4 ,
x2 − y2 = 1 .
As an initial guess, use ( x0 , y 0 )T = (1.6,1.2)T .
9.
Use Runge-Kutta fourth order formula to find y (0.4) given that
y′ =
y2 − x2
, y (0 ) = 1 .
y 2 + x2
Take the step size h = 0.2 .
10. Given y ′ = x +
y and y = 1 at x = 0 . Find approximate value of y for the range 0 ≤ x ≤ 0.4 in steps of 0 .2
by improved Euler’s method.
END