DECEMBER 2006
Code: A-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.
Euler’s
method when applied to the initial value problem 
, gives
. Then E is equal to
(A) 
. (B) 
.
(C)
. (D) 
.
b. In the integral 
, the values of the integrand are 
, and 
. Then, the
Gauss-Legendre two point formula gives
(A) 0.5434. (B) 0.5406.
(C) 0.2250. (D) 0.6536.
c. The numerical differentiation formula 
is given. Then, the error is defined
by
The value of M is
(A)
. (B)
.
(C) 
. (D)
.
d. The linear least squares polynomial approximation to the following data is
(A)
(B) 
.
(C) 
. (D) 
.
e. If 
is
the forward difference operator, then 
is equal to
(A)
. (B)
.
(C) 
. (D) 
.
f. The system of equations 
is being solved by Gauss-Jacobi
method. The iteration converges if 
is less than
(A) 2. (B) 2.5
(C) 3.0 (D) 4.0
g. The iteration method 
converges and is determining a
particular quantity. This quantity is
(A)
. (B) 
.
(C) 
. (D) 
.
h. What is the output of the following C program
main()
{
int a[ ] = { 3.8, 4.4, 5, 7.6, 6 };
int i, *s = a,*t = a;
for ( i=0; i<5; i++ )
{
printf ( "% d",*a );
++t ;
}
for ( i=0; i<5; i++ )
{
printf ( "% d",*s );
++s ;
}
}
(A) 3333334567. (B) 3333344586.
(C) 3333335576. (D) 3334444586.
i. What is the output of the following C program
main()
{
int i = 2;
switch ( i )
{
default : printf ( “hundred” );
case 1: printf ( “two hundred” );
break ;
case 2: printf ( “five hundred” );
break ;
case 3: printf ( “six hundred” );
break ;
}
}
(A) hundred. (B) two hundred.
(C) five hundred. (D) six hundred.
j. What is the output of the following C program
# define int char
main()
{
int i = 87;
printf ("sizeof ( i ) = %d", sizeof ( i ));
}
(A) 87 (B) 1
(C) 8 (D) 2
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. The smallest positive root of
the equation 
is
to be determined. Find an interval of unit length in which the root lies.
Taking the end points of this interval as initial approximations, obtain the
root correct to three decimal places using secant method. (8)
b. A method for finding 
is written as
Find the order of the method and the error constant. (8)
Q.3 a.
Solve the system
of equations 
using the Gauss-elimination method. (8)
b. The system of equations 
has a solution near x = 1.3, y
= – 1.1. Perform two iterations of the Newton’s method to improve the
solution. (8)
Q.4 a. The following system of
equations is given 
.
The system is to be solved by the Gauss-Seidel iteration method. Obtain the iteration matrix of the method. Hence, find the rate of convergence of the method. (10)
b. Find the Choleski decomposition of the symmetric matrix
(6)
Q.5 a. An equi-spaced data
with step length h is to be constructed for a function f (
x ) and approximations to function values are to be calculated by
linear interpolation from this table. Find the maximum step size that can be
used. The function values are to be correct to 
for the function 
over the interval [0, 1 ].
(8)
b. Write a C program
for finding a simple root of f ( x ) = 0 using
Newton-Raphson method. Input initial approximation to the root as 
, maximum number of
iterations as n, and error tolerance as tol. Output the value
of the root , number of iterations taken and the value of f ( root
). If iterations n is not sufficient, the program should indicate the
same. Assume that 
(8)
Q.6 a. A physicist wants to fit an
approximation of the form 
to a data. Derive the normal
equations using least squares approximation. Fit the above approximation to the
following data.
(8)
b. Construct the backward difference table for the data
Hence, compute an approximation to
. (8)
Q.7 a. A differentiation formula is defined as follows
Using the Taylor series expansions, show that 
and write the expression for 
. Use the above
formula to compute approximations to 
with step lengths h = 0.2
and h = 0.1 from the following table of values
(8)
b. The error in composite
Simpson’s rule with step length h for computing 
is bounded by 
. Assume that the
composite Simpson’s rule is being used to compute 
. Using the above error estimate find h
such that
. (8)
Q.8 a. Derive the two point Gauss-Laguerre formula of integration
. Hence, evaluate the integral 
. (6+3)
b. Write a C program for solving
an initial value problem 
using Euler’s method. Initial values
, step length h
and final value xf up to which the computations are to be carried out
are to be read. Write 
as a function sub program. Output all
the given data and all computed values. (7)
Q.9 a. Use Taylor series method of
second order to compute y(1.4) with h = 0.2, for the initial
value problem 
(8)
b. Error in Euler’s method for solving the
initial value problem 
is given by 
Euler’s method is being
used to solve the initial value problem 
Find the largest value of the step
length h that can be used such that the magnitude of the error is 
(8)