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.Evaluateusing 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 Newtonís 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 Jacobiís 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 earthís 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 Rombergís 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 Taylorís series method, solve at x = 0.1, 0.2.†††††††††††††† (12+6)††††††††††††