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 Simpsonís 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 Givenís 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 Newtonís 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)††††††††††††