Code: C-09 / T-09                                                            Subject: NUMERICAL COMPUTING

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.       The approximation  to  is correct to n significant digits.  The value of n is

   (A)  7.                                               (B)  6.

 (C)  5.                                               (D)  4.

       

b.      The equation f (x) = 0 has a simple root in the interval (1, 2).  This root is to be determined correct to two decimal places using bisection method.  The required number of iterations is    

(A)    5.                                                 (B)  6.

(C) 7.                                                  (D)  8.                                                                 

 

c.       The system of equations

     

       has                 

 

(A)    no solution.                                  

(B)    two parameter family of solutions.

(C)    one parameter family of solutions.

(D)    unique solution.

 

             d.   Gauss-Seidel iteration method is used to solve the system of equations

                  

                   The rate of convergence of this method is

 

(A)    0.7659.                                       (B) –0.7659.

(C)  1.7635.                                       (D) –1.7635.        

       

             e.   The value of  where  is the forward difference operator and E is the shift operator, with differencing h =1 is                                                      

(A)     .                                            (B)  .

(C)  6 x.                                              (D)  6

       

 

 

 

             f.    The data

                  

                   represents a third degree polynomial.  It is known that f (3) is in error.  The correct value of f (3) is 

(A)     37.                                               (B)  39.

(C)  41.                                               (D)  43.

       

             g.   The least squares polynomial approximation of degree 1 to  on [0, 1] is .  The least squares error is

(A)     .                                             (B)  .

(C)  .                                            (D)  .

             h.   The truncation error in the method

                                       

                   is of .  The value of p is

(A)    1.                                                 (B) 2.

(C)  3.                                                 (D) 4.

             i.    The minimum number of sub-intervals that will be required, so that the error in evaluating  by trapezoidal rule is , is

 

(A)   8.                                                 (B) 9.

(C) 12.                                                (D) 13.

 

             j.    The value of y(1.4) for the initial value problem  using Euler method with h=0.2 is

(A)  4.6.                                              (B)  22.5552.

(C)  25.1552.                                      (D) 26.2432.

 

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

  Q.2     a.   The method  is used to find a multiple root of multiplicity two of the equation f (x)=0.  Determine the rate of convergence of the method.  Also obtain the asymptotic error constant.                    (10)

             b.   Perform two iterations of the method   

                  

                  

                   to determine a root of  with .                                  (6)       

       

Q.3       a.   Set up the Newton-Raphson method in matrix form to solve the system of equations

                  

                  

                   Perform two iterations of the method with the initial approximation .                  (8)

                   

             b.   Solve the system of equations                                                                             

                  

                   using Gauss-Jordan method with partial pivoting, if necessary.                     (8)                                     

                                                                                                                                           

 

Q.4      a.   Using  decomposition method, find the inverse of the matrix

                   Take .                                                                      (8)

                 

 

            b.   Perform four iterations of the power method to obtain the largest eigenvalue in magnitude of the matrix .  Take the initial approximation vector as .                                                                                      (8)

 

Q.5     a.       Using Jacobi method, find all the eigenvalues and the corresponding eigenvectors of the matrix .  (Use exact arithmetic)                                                    (10)

       

          b.    Obtain the least squares approximation of the form  to the data     

                                                                (6)

         

Q.6    a.   Jacobi iteration method is used to solve the system of equations 

                  

                Determine the rate of convergence of the method.  Starting with the initial vector

                    perform two iterations.                                                         (8)

                  

 

          b.        Prove the following operator relations

               (i)    .

               (ii)   .

               (iii)  .                                                                                                               

               The operators have their standard meaning.                                                       (2+3+3)

              

Q.7   a.  Determine the largest step size h that can be used in the tabulation of a function

             f(x), , at equally spaced nodal points so that the truncation error

             of the quadratic interpolation is less than in magnitude.  Find h, when 

              and .                                                                        (8)          

        b.    For the method

              

               determine the optimal value of h, so that

               =minimum where  

               is the maximum round off error in       function values and  is the maximum value of  in the given interval.                                                 (8)

Q.8   a.   Evaluate using Simpson’s  rule with 3 and 5 nodal

               points. Find the improved value of I using Romberg integration.                              (8)                                                   

        b.    Determine the constants A, B and C in the method

                

               so that the method is of highest possible order.  Obtain the truncation error and the

               order of the method.  Use this method to obtain the value of (8)

Q.9  a.    Find the truncation error and the order of the Heun’s method for solving the initial

               value problem .                                                               (8) 

           

         b.   Using classical Runge-Kutta method of order four with h=0.1, obtain the

               approximate value of y (1.2) for the initial value problem

               .                                                                                   (8)