ALCCS

 

 

Code: CS41                                          Subject: NUMERICAL & SCIENTIFIC COMPUTING

Flowchart: Alternate Process: SEPTEMBER 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 a real root of the equation  by Regula-Falsi method correct to 4 decimal places.

 

             b.  Solve  using Gauss elimination method.

 

             c.  Find all the eigenvalues and the corresponding eigenvectors of the matrix .

 

             d.  Find the missing terms in the following table

x

1

2

3

4

5

6

7

f(x)

103.4

97.6

122.9

?

179.0

?

195.8

                 

                 

       

             e.  Obtain the least square polynomial approximation of degree two for  on [0,  1].

 

             f.   Evaluate .

                                                                                                                                                                       

             g.  Evaluate  by using the Simpson’s  rule, dividing the interval into 3 parts.  Hence find an approximate value of log .                                                                                (7  4)

 

  Q.2     a.  Solve the matrix equation

                 

                  using the Cholesky method.                                                                                              

 

             b.  Solve the equations by relaxation method   .                                                                                                                                (9+9)

                 

  Q.3     a.  Transform the matrix  to tridiagonal form by Given’s method. Use exact arithmetic.

                                

             b.  Find all the eigenvalues and the eigenvectors of the matrix A using Jacobi method where                                                                                            (8+10)

 

  Q.4     a.  Find the cubic polynomial which takes the following values using Newton’s interpolation and further evaluate f(4).

x

0

1

2

3

f(x)

1

2

1

10

                 

                                                                                                                                                          

 

 

             b.  Using Hermite interpolation determine the values of f(-0.5) and f(0.5) for the following given values of f(x) and                                                                                                                        (9+9)

x

f(x)

-1

1

-5

0

1

1

1

3

7

                                                                                                                                                          

 

 

 

 

 

  Q.5     a.  A slider in a machine moves along a fixed straight rod.  Its distance x cm along the rod is given below for various values of the time t seconds.  Find the velocity of the slider and its acceleration when t = 0.3 second.

t

0

0.1

0.2

0.3

0.4

0.5

0.6

x

30.13

31.62

32.87

33.64

33.95

33.81

33.24

                 

            

            

                 

              b.  The following table gives the temperature (in degree Celsius) of a cooling body at different instants of time t (in secs)                                                                                                                           

t

1

3

5

7

9

85.3

74.5

67.0

60.5

54.3

                 

                 

 

                  Find approximately the rate of cooling at t = 8 secs                                                    (9+9)

 

  Q.6     a.  Evaluate the integral  by subdividing the interval [0, 1] into 2 equal parts and then applying the Gauss-Legendre three point formula.                                                                                                                                                                                                                                                                

 

              b.  Evaluate the integral  using the Gauss-Hermite two point and three point formulas.                                                                                                                                (9+9)

 

  Q.7     a.  Solve the initial value problem  using the Euler method with stepsize h=0.1 to find y(0.2).                                                                                                                                        

 

             b.  Solve the initial value problemusing fourth order Runge-Kutta method on the interval [0, 0.4] with stepsize h=0.2. Compare your result with the exact solution.                      (8+10)