ALCCS

 

FEBRUARY 2009

 

Code: CS41††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Subject: NUMERICAL COMPUTING

Time: 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 †††††††††† ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (7 x 4)

†††††††††††† a.Find the relative error in the function

 

†††††††††††† b.Perform two iterations to find the fourthroot of 32, using the method offalse position.

 

†††††††††††† c.Factorize the matrix using LU decomposition.

 

†††††††††††† d.Determine the largest eigenvalue in the fourth approximation and its corresponding eigenvector of the matrix using Power method.

 

†††††††††††† e.Apply Gauss Jordan method to solve the equations AX=B where ,

 

†††††††††††† f.†† Find

 

†††††††††††† g. Solve at 0.1 using Euler method.

 

Q.2†††††† a.Find the root of xex=3 by Regular falsi method correct to three decimal places.††††††††††††† (9)

†††††††††††††††††††††††††††††††††††

†††††††††††† b.Find the missing values in the following table of values of x and y:†††††††††††††††††††††††††††††††††† (9)

x

0

1

2

3

4

5

6

y

-4

-2

----

---

220

546

1148

†††††††††††††††††

 

 

 

Q.3†††† a.Find the inverse of by Croutís method.††††††††††††††††††††††††††††††††††††††††††††††† (9)

 

†††††††††††† b.Using Givenís Method, reduce the following matrix to the tri-diagonal form:††††††††††††††††††† (9)

††††††††††††††††† A=

 

Q.4†††† a.Solve by Gauss elimination method, the following system of equations:††††††††††††††††††††††††††† (9)

†††††††††††††††††

 

†††††††††††† b.Determine the order of convergence of the iterative method for finding a simple root of the equation f(x)=0.†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (9)

 

Q.5†††† a.The following table gives the values of density of saturated water for various temperatures of saturated steam.††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 

T=temp0 C

100

150

200

250

300

d = density (hg/m3)

958

917

865

799

712

†††††††††††††††††

 

 

††††††††††††

††††††††††††††††† Find by Newtonís divided difference interpolation the densities when temperature are 1300C and 2750C respectively.††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (9)

 

††††††† †††† b.Use Lagrangeís interpolation formula to find the value of y when x = 10, if the values of x and y are given as below:††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (9)

†††††††††††††††††

x

5

6

9

11

y

12

13

14

16

††††††

Q.6†††† a.The population of a certain town is shown in the following data:††††††††††††††††††††††††††††††††††††† (9)

†††††††††††††††††

Year

1951

1961

1971

1981

1991

Population (in thousands)

19.96

36.65

58.81

77.21

94.61

††††††††††††

†††††††††††† †††† Find the rate of growth of the population in the year 1981, using Newtonís difference formula.

 

 

 

 

†††††††††††† b.The velocity v of a particle at a distance s from a point on its path is given by the following table:††††† (9)

†††††††††††††††††

s(ft)

0

10

20

30

40

50

60

v (ft/s)

47

58

64

65

61

52

38

 

††††††††††††††††† Estimate the time taken to travel 60 ft using Simpsonís 1/3 rule.

 

Q.7†††† a.Using Runge-Kutta methodof fourth order, solve for y(0.1), y(0.2) given that.††††† (12)

 

†††††††††††† b.Using Taylorís series method, solve at x = 0.1, 0.2.†††††††††††††††††††† (6)