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

July 21, 2018 Lokesh Kumar 106 Views 0 Comments 2nd Year, BSc, Computer Science, Exam paper

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) ?

(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 $y_{x}$ .

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