DECEMBER 2006

## 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)