## 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 second degree Taylor series polynomial expansion of the function  about x = 0 is

(A)  .                       (B)  .

(C)  .                            (D)  .

b.      A root of the equation lies in the interval (0.5, 1.0).  The number of iterations required by the bisection method to obtain this root correct to 2 decimal places is

(A)    13.                                               (B)  11.

(C) 9.                                                  (D)  7.

c.       Newton-Raphson method is used to obtain the approximate value of .  If the initial approximation is taken as , this the value obtained after two iterations is

(A)    2.5.                                              (B)  2.25.

(C)  2.2142.                                        (D)  2.1242.

d.   The homogeneous system of equations

has

(A)    two parameter family of solutions.

(B)    one parameter family of solutions.

(C)  only zero solution.

(D)  no solution.

e.   The spectral radius of the Gauss-Jacobi iteration method to solve the linear system of equations

is

(A)     1.26.                                            (B)  1.50.

(C)  2.00.                                            (D)  2.50.

f.    The data

represents an nth degree polynomial.  The value of n is

(A)     4.                                                 (B)  3.

(C)  2.                                                 (D)  1.

g.   The truncation error in the method

is

(A)     .                                    (B)  .

(C)  .                                 (D)  .

h.   The value of the integral

using two point Gauss-Hermite method is

(A)    0.5908.                                        (B) 1.1813.

(C)  2.0944.                                        (D) 4.1888.

i.    The value of b in the formula

is

(A)   1.                                                 (B) 2.

(C) 4.                                                  (D) 8.

j.    The approximate value of y(1.4) for the initial value problem

y(1) = 2

obtained by using the Euler’s method with h=0.2 is

(A)  2.5472.                                        (B)  2.7596.

(C)  2.7637.                                        (D) 2.9253.

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Derive the secant method to determine a simple root of the equation f(x) = 0.  Find its rate of convergence.                                                            (8)

b.   Find an interval of unit length which contains the smallest negative root in magnitude of the equation

(i)                  Perform two iterations of the bisection method.

(ii)                Taking end points of the last interval as initial approximations, perform two iterations of the secant method.            (8)

Q.3       a.   Perform two iterations of the Newton’s method to solve the system of equations

Take the initial approximation as (0.8, 0.9).                                                      (10)

b.   Find all solutions of the system of equations

using Gauss-elimination method.                                                                       (6)

Q.4      a.     Set up the Gauss-Seidel iteration method in matrix form to solve the linear system of equations

Find its rate of convergence, if the method is convergent.                      (8)

b.    Find the inverse of the matrix

using Choleski method.                                                                                (8)

Q.5     a.       Using five iterations of the inverse power method, find the eigenvalue which is nearest to 4 in magnitude for the matrix.

Take the initial approximate eigenvector as .                            (8)

b.      Using Given’s method, reduce the matrix

to tri-diagonal form.  Obtain the Sturm’s sequence and hence all the eigenvalues.         (8)

Q.6    a.   Prove the following operator relations

(i)

(ii)                                                                                    (4)

b.      Using Newtonts divided difference interpolation, obtain the polynomial

which fits the data

(6)

c.   Obtain the least squares approximation of the form  for the data

(6)

Q.7   a.  Obtain the maximum absolute truncation error  and the maximum

absolute round off error  in the method

where

(i)    for all i

(ii)   is the maximum round off error in evaluating y(x) for all x

(iii)  for all x in the given interval.

Determine the optimal value of h so that .                                                (8)

b.    Obtain the Richardson’s extrapolation scheme for the method

Using this method and Richardson’s extrapolation, find the best value of

using the data

(8)

Q.8   a.  Find the minimum number of intervals required to evaluate

using the Simpson’s rule with an accuracy of .                         (8)

b.    Obtain the values of the constants  and  in the method

where  is the weight function, so that the method is of highest possible

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

Q.9   a.      Solve the initial value problem

in the interval [1, 1.4] using the classical fourth order Runge-Kutta method with h=0.2.                                       (8)

b.   Using the multi-step method

with h = 0.2, obtain the approximate value of y(1.8) for the initial value problem

Find the starting values using the Euler method with h = 0.2                                         (8)