Code: A-07               Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING

Time: 3 Hours                                   June 2006                                                              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. An integral is being evaluated by the three point trapezoidal rule as

where  The round off errors in      and  Then, the round off error in evaluating the integral is bounded by

(A) . (B) .

(C) . (D)  .

b. A negative root of the equation  lies in the interval

(A) . (B) .

(C) . (D) .

             c.   We want to use the Gauss - Jacobi iteration method to solve the system

                   Let  be the approximate initial solution vector. Then, the iteration

(A) converges for all . (B) diverges for any .

(C) has rate of convergence 1.35. (D) has rate of convergence 1.

             d.   The backward difference  is approximately equal to

(A) .                               (B) .

(C) .                               (D) .

             e.   The polynomial that fits the data

                   is

(A) . (B)  .

(C)  . (D) .

             f.    The following data for a function f (x ) is given.

Then, applying the trapezoidal rule with suitable step length, we obtain the value of the integral  as

(A) 105. (B) 115.

(C) 75. (D) 145.

             g.   The numerical differentiation formula

                   is given. Then, the order of the formula is

(A) 1.                                                 (B) 2.

(C) 3.                                                 (D) 4.

             h.   Taylor series method of second order with step length h = 0. 1 is used to approximate for the initial value problem . The approximate value of is

(A) 2.25. (B) 2.65.

(C) 3.2. (D) 2.32.

             i.    Find the output for the following C program

                   #include<stdio.h>
main()
{
char S1[]="Butter";
char S2[]="Milk";
S1=S2;
printf("%s",S1);
}

(A) Milk.                                             (B)  Butter.

(C) Butter Milk. (D) Compilation error saying 'lvalue

required'.

             j.    Find the output for the following C program

                   main()
{
int x = 5;
printf("%d %d %d\n", x, x<<2, x>>2);
}

(A)  5  5  5. (B) 5  20  1.

(C) 5  3  7. (D) Compilation error.

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Locate a root of the equation in an interval of length one unit.

(i) Perform two iterations of the bisection method to improve the approximation to the root. (ii) Taking the mid point of the last interval obtained in (i) as initial approximation perform two iterations of the Newton-Raphson method.                                                                                               (8)

     b.   Define the order and error constant of an iterative method for finding a root of the equation . An approximation for finding  is being computed by the iteration method

Find the order of the method and the error constant. (8)

Q.3     a.   The system of equations  has a solution near x = 1.3, . Perform two iterations of the Newton’s method to improve the solution. (8)

             b.   Solve the system of equations 

by the Choleski method.                                                                                    (8)

Q.4     a.   Using Gauss elimination, find the conditions under which the system of equations

                   has a unique solution or no solution.                                                                    (6)

b.   The system of equations

is to be solved by the Gauss-Jacobi iteration method. Obtain the iteration matrix of the method. Find the rate of convergence of the method. (Use the Newton-Raphson method to find a root of the characteristic equation of the iteration matrix). (10)

Q.5     a.   Fit a polynomial for the following data

(5)

     b.   If , evaluate, where is the backward difference operator.                (4)

c. Write a C – program for evaluating the Lagrange polynomial of degree N based on N + 1 points  Input maximum degree as 10.                                                  (7)

  Q.6     a. Use Newton’s backward difference polynomial to interpolate at x = 1.5, from the     following data

(7)

             b.   A physicist studying a decaying process decides to fit an approximation of the form to a data. Derive the normal equations to determine a and b using the least squares approximation. Fit the above approximation to the following data

(9)

Q.7     a.   A numerical differentiation formula is defined as

                   Find the values of a, b, c such that the formula is of as high order as possible. Find the leading term of the truncation error.                                   (8)

b.   Evaluate   using Simpson’s rule with 2 and 4 subintervals. Compare these values with the exact solution. (8)

Q.8     a.   Consider the numerical integration formula

.

Find the values of the parameters a, b,  such that the formula is exact for polynomials of order as high as possible. Find the error term.               (8)

             b.   Write a C – program to evaluate the integral   by trapezoidal rule with N equal subintervals, using the function as  (8)

Q.9     a.   Evaluate the integral  using the Gauss-Legendre three point formula.             (6)

b.   Derive the truncation error term of the Euler method for solving the initial value problem .                   (4)

c.   Use the classical Runge-Kutta method of fourth order to find the numerical solution at for the initial value problem

with .                                                                                                     (6)