ALCCS

 

 

Code: CS41                                          Subject: NUMERICAL & SCIENTIFIC COMPUTING

Flowchart: Alternate Process: MARCH 2010Time: 3 Hours                                                                                                     Max. Marks: 100

 

NOTE:

·      Question 1 is compulsory and carries 28 marks. Answer any FOUR questions from the rest.  Marks are indicated against each question.

·      Parts of a question should be answered at the same place.

·      All calculations should be up to three places of decimals.

 

 

  Q.1     a.  Find the number of terms of the exponential series such that their sum gives the value of ex correct to six decimal places at x = 1.

 

             b.  Find a root of the equation, by the secant method upto two iterations.

 

             c.  Factorize the matrix  using LU decomposition.

 

             d.  Given the matrix A = I + L + U, where , L and U are the lower and upper triangular matrices respectively,  decide whether Jacobi method converges to the solution of Ax = b.

                 

             e.  Evaluate using Gauss formula for n = 2.

 

             f.   Solve

                  at x = 0.1 using Euler method.

                 

             g.  Show that  .                                                                                       (7  4)

 

  Q.2     a.  Find a real root of   correct to four decimal places using Newton’s Method.          

 

             b.  One entry in the following table is incorrect and y is a cubic polynomial in x. Use the difference table to locate and correct the error:                                                                                                (9+9)

x

0

1

2

3

4

5

6

7

y

25

21

18

18

27

45

76

123

                 

            

 

 

  Q.3     a.  Solve by Gauss-Seidel method, the following system of equations:

                                                                                                                             

 

             b.  Solve by Relaxation method, the system of equations:

                                                                                                                (9+9)

 

  Q.4     a.  Using Jacobi’s method, find all the eigenvalues and the eigenvectors of the matrix

                                                                                                                        

 

             b.  Using inverse interpolation, find the real root of the equation  which is close to 1.2.     (9+9)

 

  Q.5     a.  The following table gives the distance in nautical miles of the visible horizon for the given heights in feet above the earth’s surface :

x = height

100

150

200

250

300

350

400

y = distance

10.63

13.03

15.04

16.81

18.42

19.90

21.27

                 

            

            

                  Find the value of y when x =  218 ft and x = 410 ft.                                                           

 

              b.  From the given data, find the maximum value of y:                                                      (9+9)

x

-1

1

2

3

y

-21

15

12

3

                 

                 

 

 

  Q.6     a.  Use Romberg’s method to compute correct to four decimal places.                                                                                                                                                                                                     

              b.  Determine f(x) as a polynomial in x for the following data:                                          (9+9)

 

x

-4

-1

0

2

5

y

1245

33

5

9

1335

                 

 

 

  Q.7     a.  Using Runge-Kutta method of fourth order, solve for y(0.1), y(0.2) given that  .      

 

             b.  Using Taylor’s series method, solve at x = 0.1, 0.2.               (12+6)