AMIETE – CS/IT (OLD SCHEME)

 

Code: AC09 / AT09                                                         Subject: NUMERICAL COMPUTING

Flowchart: Alternate Process: DECEMBER 2009Time: 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 the best alternative in the following:                                   (2x10)

 

a.       The true value of a number is 20. It is approximated as 19.995. Then, the percentage relative error in the approximation is given by 

                                                                                                                                             

  (A)  0.005                                            (B)  0.02.

                    (C)  0.025.                                           (D)  0.05.

       

b.      Newton-Raphson method is applied to approximate the value of  The formula is given by

 

(A)                                 (B) 

(C)                                    (D)                                               

 

c.       The Cholesky factorisation of a symmetric matrix is  If

                                     ,  and    ,  c > 0

                   then, the values of (a, b, c) are

(A)    (1, 1, 1).                                      (B)  (-1, -1, 0).

(C)  (-1, -2, 1).                                  (D)  (-1, -1, 1).

 

d.      The matrix is to be reduced to the tri-diagonal form by the Givens method. The angle  of the orthogonal rotation is given by

 

(A)                                              (B) 

(C)                                            (D)    

 

e.       The Newton’s interpolation polynomial that fits the data

 

                                      

                   is given by

(A)                                    (B) 

(C)                                    (D) 

 

f.    The numerical differentiation formula                   is given. The error in the formula is written as  The value of p is

            

                    (A)  1                                                  (B)  2

                    (C)  3                                                  (D)  4

 

             g.   The value of the integral  obtained by the Gauss-Chebyshev two point formula is given by

 

(A)                                  (B) 

 (C)                        (D) 

 

             h.   The initial value problem  is solved by the Taylor series method of second order. Then, the approximate value of y(2.1), with h = 0.1 is given by

 

(A)    1.25                                             (B)  1.5.

(C)  1.225                                           (D)  2.225.

 

i.      The following data is given.

                  

        The least squares linear polynomial approximation to the above data is      

 

(A)                                            (B) 

(C)                                           (D) 

 

             j.    The iterative method   where N is a positive number, is being used to evaluate a certain quantity. If the iteration converges, the method is finding the value of

 

(A)                                             (B) 

(C)                                              (D) 

 

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

 

 

  Q.2     a.   The real root of the equation  is to be determined.      (8)

                   (i)   Find an interval of length 1 in which the root lies.

                   (ii) Starting with this interval, perform two iterations of the bisection method.

                   (iii) Taking the end points of the last interval obtained in (ii) as initial approximations, perform two iterations of the Regula-Falsi method.

 

             b.   Find the rate of convergence of the Newton-Raphson method  for finding a simple root of the equation .                    (8)

                                                                             

  Q.3     a.   Set up the Gauss-Seidel iteration scheme in matrix form to solve the linear system of equations                                                                   (8)

                        

                   Determine the rate of convergence of the method.

                                                                                                                                     

             b.   Solve the following system of equations by LU decomposition method with  

                                                          (8)

 

Q.4      a.     Using the Jacobi method, find all the eigen values and the corresponding eigen vectors of the matrix

                                           

                   Use exact arithmetic.                                                                                          (8)

           

            b.    Estimate the value of the largest eigen value in magnitude and the corresponding eigen vector of the matrix   (8)

                                            

                   by the power method correct to two decimal places. Assume the initial eigen vector as  [1,   0.02,   1   

 

Q.5      a.   The method   is known as a second order formula. Compute estimates for  using all possible values of step lengths from the given table of values and improve these estimates using Richardson’s extrapolation.                                   (8)

                                   

               

b.  The function defined by the following table of values has a minimum. Find the value of x at which  f(x) attains the minimum and find the minimum  value.                                                             (8)

                                    

 

  Q.6     a.  A table of values is to be constructed for the function  in the interval [0, 1]. It is proposed to use linear interpolation on this interval. Find the largest step length h that can be used in order that error in linear interpolation is less than                                                                (8)

 

             b.  Obtain the least squares straight line fit to the following data:

 

                       x:          0.2          0.4          0.6          0.8          1

                       f(x):      0.447      0.632      0.775      0.894      1                                            (8)

 

  Q.7     a. Evaluate the integral  using the Gauss-Legendre three point   formula.             (8)

                 

             b.  Compute the integral   using Simpson’s rule with 3 and 5 points.   Improve the results using Romberg integration                              (8)

 

  Q.8   a.    Use the Taylor series method of second order to integrate the initial value problem  with h = 0.1.                                                                               (8)

                 

            b.   Find an approximation to y(0.1), using the fourth order Runge-Kutta method for the initial value problem            (8)

                                    with   h = 0.1.

Q.9      a.   The integration formula   is given. Find the values of a and c such that the formula integrates exactly                                                                   (6)

            b.   If , where  is the forward difference, find the values of p and q.                             (5)

 

            c.   The initial value problem  is given. Find the approximation to y(0.2) using Euler method with h = 0.1.                                                                                     (5)