AMIETE – ET (OLD SCHEME)
Code: AE07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING
Time: 3 Hours Max. Marks: 100
NOTE: There are 9 Questions in all.
· Question 1 is compulsory and
carries 20 marks. Answer to Q.1 must be written in the space provided for it in
the answer book supplied and nowhere else.
· Out of the remaining EIGHT
Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not
explicitly given, may be suitably assumed and stated.
Q.1 Choose
the correct or the best alternative in the following: (2
10)
a. The value of 
is approximated by
22/7. If seven significant digits are used,
the percentage relative error in this approximation is
(A) 0.4. (B) 0.04.
(C) 2.2. (D) 4.
b. A root of the equation 
is near 
Using one iteration of
the Newton-Raphson method, the next approximation to
the root is obtained as
(A) -1.25115. (B) -1.29856.
(C) -1.70161. (D) 1.26211.
c. The data P(1) = 14, P(2) =35, P(3) = 72 is given. An approximation to P(2.5) using all the data values is given by
(A)
61.5 (B) 53.5
(C) 56.0 (D) 51.5
d. Let 
be a third degree
polynomial. Then, the forward difference 
is given by
(A) 6a. (B) 0.
(C) 3a. (D) 6a + b.
e. The value of the integral 
evaluated by the
trapezoidal rule with h = 1, is
obtained as
(A)
16.292 (B) 18.292
(C) 15.322
(D)
16.992
f. The initial value problem 
y(0) = 1 is given. The
approximation to y(0.4) obtained by
the Euler method with h = 0.2 is
(A)
1.0.
(B)
0.62.
(C) 0.82. (D) 0.92.
g. If 
is an eigen value of
A, then eigen value of A-1 is
(A) 
(B) -
(C) - (D) 
h. If
f(x) = 
, then the value of f[a,b] will be
(A) 
(B) 
(C) 
(D) 
i. For the Simpson’s 
rd rule, the interpolating polynomial is a
(A)
Straight-line (B)
Parabola
(C) Cubic curve (D) None
j. The
value of the integral 
using Gaussian integration formula for n = 2 is
(A) 
(B) 
(C) 
(D) None
Answer any FIVE Questions out
of EIGHT Questions.
Each
question carries 16 marks.
Q.2a. A root of the
equation 
is to be
determined. Obtain an interval of length 1 unit, in which a positive root lies.
Taking the end points of this interval as initial approximations, perform two
iterations of the secant method to find approximations to the root (7)
b. Write a C
program to compute 
using while loop.
Output x, n and . (6)
c. If 
, then find the values of a
and b. (3)
Q.3a. Perform four
iterations of the Newton-Raphson method to find the smallest positive root of
the equation
f(x)=x3-5x+1=0
(8)
b. Using Lagrange interpolation, find 
from the table of
values
(8)
Q.4a. Derive the least squares straight line approximation 
for a data of N values 
Hence, obtain the
least squares straight line approximation to the data
Also, find the least squares error. (8)
b. Write a C
program to find a simple root of 
by the regula-falsi
method. Input (i) a, b (two initial approximations between
which the root lies), (ii) n (maximum
number of iterations) and (iii) error tolerance “eps”. Output (i) approximate
root, (ii) number of iterations taken. If the inputted value of n is not sufficient, the program should
write “Iterations are not sufficient”. Write the subprogram for
as
. (8)
Q.5a. The following data represents the function 
Estimate 
using the Newton’s
backward difference interpolation. Find the magnitude of the actual error. (8)
b. The system of equations 
has a solution near x = 0.7, y = 0.3. Perform two iterations of the Newton’s method to obtain
the root. (8)
Q.6a. Find the inverse of the coefficient matrix of the system of
equations
by the Gauss-Jordan method with
partial pivoting and hence solve the system. (8)
b. The system of equations
is given. Using the Gauss-Seidel
iteration scheme in matrix form for its solution, find whether the scheme
converges. If it converges, find the rate of convergence. (8)
Q.7a. The following data is given
x 1.0 1.5 2.0
2.5 3.0
f(x) 3.0 6.625 13.0
22.875 32.5
From this data, evaluate 
using the Simpson rule
with three and five points. (7)
b. Write a C
program to evaluate 
, by trapezium rule of integration based on n + 1 points. Input the values of n, a,
b. Write 
as a function sub- program. Output all the data and the
computed value. (9)
Q.8 a. The formula
is suitable for
approximating 
where a is the last value in the data.
Calculate 
from the table of
values, using all possible step lengths.
x
1.6 1.8 1.9 1.95 2.0
f(x)
7.5530 8.8497 9.5859 9.9787 10.3891 (8)
b. The formula 
is being used to
compute 
from a table of
values. Using Taylor series, find the leading term of the error. Derive the
formula for obtaining the improved (Richardson’s) extrapolated value of
. In a particular problem, the following results were
obtained with different step lengths, using the above formula.
h 0.4 0.2
0.52601 0.53671
Compute the
improved (Richardson’s) extrapolated value of. (8)
Q.9a.
Gauss-Legendre two point integration formula can be written as
p 
Determine the values of b, d, p. (7)
b. Using the Gauss-Laguerre two point formula,
evaluate the integral 
(4)
c. Use Runge-Kutta method of fourth order to
determine y(0.2) with h = 0.2, for the initial value problem 
(5)