Code: AC-09 / AT-09                                                       Subject: NUMERICAL COMPUTING

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 root of the equation  is obtained by using the Newton-Raphson method with initial approximation .  The root obtained after one iteration is  

   (A)  3.5151. (B)  3.5306.

 (C)  4.5151.                                      (D)  4.5306.

b. The rate of convergence of the secant method to find a simple root of  is m and that of the Regula-Falsi method is n.  Then   

(A) m < n. (B)  m > n.

(C) m = n. (D) m = 2n.

c. The rate of convergence of the Gauss-Jacobi method for solving the system of equations

                                                                        is                    

(A) 0.98.                                            (B)  0.85.

(C)  0.56.                                            (D)  0.49.

             d.   Given’s method is used to reduce the matrix  to tri-diagonal form.  The angle of orthogonal transformation is

(A) .                              (B) .

(C) .                                              (D)  .    

             e.   The interpolating polynomial which fits the data

                   is                                                              

(A) . (B)  .

(C)  .                                  (D) .

f.    The operator  is equivalent to the operator                               

(A) . (B) .

(C) . (D) .

             g.   The least squares polynomial approximation of degree 1 to the function  on  is a + bx.  The value of b is                                                              

(A) . (B) .

(C) . (D) .

             h.   The truncation error of the method

                   is

(A) . (B) .

(C)  . (D) .

             i.    The value of the integral  using Gauss-Legendre two-point method is

(A) . (B) .

(C) . (D) .

             j.    The approximate value of y(1.2) for the initial value problem

                   obtained by using the Taylor series second order method with h=0.2 is 

(A)  2.84. (B) 2.65.

(C) 1.84. (D) 1.65.

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2     a.   Perform three iterations of the Regula-Falsi method to obtain the approximate value of  which lies between 3 and 3.2.  To how many decimal digits is this value accurate?                              (8)

             b.   Find the rate of convergence and the asymptotic error constant for the method

                   to obtain a simple root of the equation .                                                (8)

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

                   Take the initial approximation as .                                         (9)

             b.   Using LU decomposition method solve the system of equations                                                                                                 

(7)

Q.4      a.     Obtain the inverse of the matrix

                   Using Gauss-Jordan method.  Hence solve the system of equations where .                                                                                                                          (9)

b.    Show that the Gauss-Jacobi iteration method for solving the system of equations

                   is divergent.                                                                                                     (7)

Q.5     a.       Find all the eigenvalues and the corresponding eigenvectors of the matrix

                   Using Jacobi’s method.                                                                                   (8)

b.      Find the least squares approximation of the form  to  on the interval .                                                                  (8)

Q.6    a. Prove the following operator relations

                     (i)  

                     (ii)  (4)

b. Using Lagrange interpolation and the data

    obtain the interpolating polynomial which fits this data.  Hence obtain an     

      approximate value of .                                                                         (6)

             c.   Obtain second degree Taylor series polynomial approximation to the function  about x = 2.  Find the maximum absolute error if this approximation is used in the interval .       (6)

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

              absolute round off error  in the method

.  Assuming that  is the maximum round   

     off error in evaluating f(x) and , where , 

     determine the optimal step size h so that .                            (8)

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

               Using this method and the corresponding Richardson’s extrapolation, obtain the   

               best value of  using the data

(8)

Q.8   a.  Evaluate the integral  using Simpson’s rule of integration with 3 and 5 nodal points.  Find the best value using Romberg integration.                                                                       (8)

b.    Obtain the values of the constants  and  so that the method

               is of highest possible order.  Determine the order and error term of the method. 

               Use this method to find the value of .                                         (8)

Q.9   a.      Obtain the order and truncation error of the method

for solving the initial value problem .                                     (8)

b.   Using the classical fourth order Runge-Kutta method with h = 0.2, obtain the

               approximate value of y (0.4) for the initial value problem. (8)