Code: AE-07 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 best alternative in the following: (2x10)
a.
The
initial value problem 
is given. An approximation to 
by Taylor series method of second order with h = 0.2, is given by
(A) 1.22. (B) 1.2.
(C) 0.98. (D) 1.42.
b.
The integration formula 
is exact for 
. The values of a,
b respectively are
(A)
. (B)
.
(C) 
. (D)
.
c. In the numerical differentiation formula
, the value of a
is
(A) 3. (B) –1.
(C) –3. (D) 1.
d. The linear least squares approximation to the following data is
(A)
. (B) 
.
(C) 
. (D) 
.
e. The polynomial p(x) of degree 2 satisfying the data 
is
(A)
. (B)
.
(C) 
. (D) 
.
f. The first divided difference 
is approximately equal to
(A)
. (B)
.
(C) 
. (D)
.
g. What is the output of the following C program
#include<stdio.h>
main()
{
int a, b = 1, c = 3 ;
a = 2*b + c;
b = 2*a + b/3;
c /= 2;
a = b + c + a ;
printf ("\n b = % d \ c = % d \ a = % d \ n", b, c, a );
}
(A)
. (B) 
.
(C) 
. (D) 
.
h. The body of a C program is as follows
main()
{
int sum = 0, digit;
long number, a;
clrscr();
printf ( "\nEnter positive integer number\n " );
scanf (" %ld", &number );
a = number;
while (a>0)
{
digit = a%10;
a/= 10;
sum+= digit;
}
printf ( "\n Result of %ld = %d \n", number, sum );
}
If the input is 5276, then output is : Result of 5276 = ? . The value of ? is
(A) 16. (B) 20.
(C) 26. (D) 52.
i. If the if statement does not have associated else, then what happens if the condition is false.
(A) gives an error message.
(B) Control is passed to the beginning of the program.
(C) Control is passed to the next statement after the if statement.
(D) Control is passed to return.
j. A root of 
is being
determined by the Regula-Falsi method. It was found that it lies in the
interval (1.8385, 2 ). In the next iteration, the root lies in
(A) (1.8385, 1.8536). (B) (1.8465, 2).
(C) (1.8536, 2). (D) (1.8465, 1.8536).
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. A positive root of the
equation 
lies
in the interval (2, 3).
(i) Starting with this interval, apply bisection method two times.
(ii) Taking the mid-point of the last interval obtained in (i) as initial approximation, refine the root using two iterations of Newton-Raphson method. (9)
b. Find the rate of convergence
of the secant method for finding a simple root of the equation 
. (7)
Q.3 a.
Find the
inverse of the matrix 
using the LU decomposition method. (8)
b. The system of equations 
has one solution near 
. Obtain the
approximation to the solution using two iterations of Newton’s method. (8)
Q.4 a. Perform three iterations of Gauss-Jacobi method for the solution of the system of equations
taking the initial
approximations as 
. (6)
b. Obtain the iteration matrix of the Gauss-Jacobi method for the problem in (a). Hence, find the rate of convergence of the method. (10)
Q.5 a. Use divided differences to fit a polynomial for the data
(6)
b. If 
, then find the values of a and
b. (3)
c. Write a C program for finding a simple root of using Regula-Falsi method.
Input the end points of the interval (a, b) in which the root lies,
maximum number of iterations allowed (n), and the error tolerance tol.
Output the value of the root and number of iterations
taken. If number of iterations n is not sufficient, the program should
output the same. Assume that 
(7)
Q.6 a. Construct the forward difference table for the data
and hence find the value of f (0.3) (7)
b. A given data is to be approximated by the quadratic polynomial 
. Derive the
normal equations using the least squares approximation. Hence, find the
approximation to the data
(9)
Q.7 a. Define a differentiation rule
as 
. Using
Taylor series expansions, show that 
and write the expressions for 
. Using the above
formula, compute approximations to 
with step lengths h = 0.2 and
0.1 from the following table of values.
(9)
b. The error in the composite
trapezoidal rule for evaluating 
is bounded by 
Composite trapezoidal
rule is being used to compute 
. Using the above error estimate, find
h such that
. (7)
Q.8 a. Write a C program to evaluate 
by Simpson’s
rule of integration based on 
points. Input the values of the
limits a, b and n. Write 
as a function program. Output all
the data and the computed value. (7)
b. Evaluate the integral 
using Simpson’s
rule with 2, 4 and 8 subintervals. Hence, obtain a better estimate using
Romberg integration (extrapolation).
(9)
Q.9 a.
Evaluate the
integral 
using
the Gauss-Legendre two point formula. (8)
b. Given the initial value problem 
estimate y(1.4)
with h = 0.2, using the Heun’s method. (8)