Numerical Analysis 2016 – BSc Computer Science Part 2 (MJPRU)

Paper code: 13512
1512
B.Sc. (Computer Science) (Part 2)
Examination, 2016
Paper No. 1.3
NUMERICAL ANALYSIS

Time: Three Hours] [Maximum Marks: 50

 

Note: Attempt five questions. All questions carry equal marks. Symbols are as usual use or calculator is allowed.

1. (a) Show that:

 \sum_{k=0}^{n-1} \Delta^{2}f_{k} = \Delta f_{n}-\Delta f_{0}

(b) Estimate the missing term in the following table:

X

F(x)

0

1

1

3

2

9

3

?

4

81

 

Explain why value differs from 3^{3} or 27?

2. (a) Given:

 u_{0} + u_{8} = 1.9243, u_{8} +u_{7}=1.9590
 u_{2} + u_{6} = 1.9823, u_{3} +u_{5}=1.9956

Find u_{n}.

(b) Use the method of separation of symbols to prove that:

 u_{1}x + u_{2}x^{2}+u_{3}x^{3}+........=\frac{x}{1-x}u_{1}+\frac{x^{2}}{\begin{pmatrix}1-x\end{pmatrix}^{2}}\Delta u_{1}+\frac{x^{3}}{\begin{pmatrix}1-x\end{pmatrix}^{3}}\Delta^{2} u_{1}+.....

3. (a) State and prove Newton-Gregory formula for forward interpolation.

(b)f\begin{pmatrix}x\end{pmatrix} = \frac{1}{x^{2}} , find the divided differences:

f(a,b), f(a,b,c) And f(a,b,c,d) ?

  1. (a) Find  \intop\nolimits_{0}^{1} \frac{dx}{1+x^{2}} by using Simpson’s 1/3 and 3/8 Hence obtain the approximate value of  in each case.

(b) Find first and second derivatives of the function given below at the point x=1.2:

x y
1 0
2 1
3 5
4 6
5 8

5. (a) Show that the expirations given below are approximations to the third derivative of y_{x}.

  1.  \Delta^{3}y_{0}+ \begin{pmatrix}x-\frac{3}{2}\end{pmatrix}\Delta ^{4} y_{0}
  2.  \Delta ^{3}y_{-1}+ \begin{pmatrix}x-\frac{1}{2}\end{pmatrix}.\frac{1}{2} \begin{pmatrix}\Delta ^{4}y_{-2}+\Delta ^{4}y_{-1}\end{pmatrix}

(b) Define the following:

  1. Inherent errors
  2. Round-off errors
  3. Truncation errors

6. (a) Solve the following system of equations by Gaussian elimination method:

 5x-2y+z=4
 7x+y-5z=8
 3x+7y+4z=10

(b) Find the solutions of the system:

 83x+11y-4z=95
 7x+52y+13z=104
 3x+8y+29z=71

7. Tabulate by Milne’s method the numerical solution of \frac{dy}{dx}=x+y with x_{0}=0, y_{0}=1, from x=0.20 to x=0.30.

8. (a) Find the real root of the equation x^{2}+4\sin x=0 correct to four places of decimals by Newton-Rap son method.

(b) Show that the square root of N=AB is given by  \sqrt{N} = \frac{S}{4}+\frac{N}{5} where S=A+B.

9. (a) Determine the real root of \tan x = x by iteration method.

(b) Use Runge-Kutta Method to approximate y, when x=0.1 and x=0.2 and , given that x=0, y=1 and \frac{dy}{dx}=x+y.

……..End……..

Lokesh Kumar

Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

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