Tribhuvan University

Institute of Science and Technology

Model Set Ii

Bachelor Level / third-semester / Science

Computer Science and Information Technology( CSC212 )

Numerical Method

Full Marks: 60 + 20 + 20

Pass Marks: 24 + 8 + 8

Time: 3 Hours

Candidates are required to give their answers in their own words as far as practicable.

The figures in the margin indicate full marks.

SECTION A

Attempt any TWO question.

1

What do you mean by ill condition? Compare Gauss elimination method and Gauss jordan method of solving simultaneous equations.

Using Gauss jordan method solve the given system of equation

6x1 – 2x2 + x3 = 11

x1 + 2x2 – 5x3 = -1

-2x1 + 7x2 + 2x3 = 5

2

Define the terms interpolation and extrapolation. Find the Langrange interpolation polynomial to fit the following data  and find value of y(10).

 x 5 6 9 11 y 12 13 14 16

3

Solve the ordinary differential equation given below by using Eular,s method. And calculate the value of y’ = 3x+ 1 ;  y(0) = 2, when

1. h = 0.5
2. h = 0.25

SECTION B

Attempt any EIGHT question.

4

Write down the program for solving ordinary differential equation using Heun’s method.

5

Find the missing term in the following using Newton’s divided difference formula.

 x 0 1 2 3 4 y 1 3 9 …. 81
6

Using a method of least square find the relation of the form y = ax+b

 x 0.301 0.4771 0.6021 0.699 y 1.444 1.7931 2.0414 2.2068

7

Find the largest eigen value and the corresponding eigen vector of the following matrix :

$$\begin{bmatrix}1 & 2 & 0\\ 2 & 1 & 0\\ 0 & 0 & -1\end{bmatrix}$$.

8

Factorize the matrix using Cholesky’s decomposition.

$$\begin{bmatrix}1 & 2 & 3\\ 2 & 8 & 22\\ 3 & 22 & 82\end{bmatrix}$$.

9

Use Romberg estimate to evaluate R(2,2)

$$∫\frac{{1}}{{1+x}}$$ from 0 to 2.

10

Calculate the integral value of the following tabulated function from x = 0 to x = 1.6 using simpson’s 3/8 rule.

 x 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 f(x) 0 0.24 0.55 0.92 1.63 1.84 2.37 2.95 3.56
11

Solve the Poison’s equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.

12

Estimate a real root of the following non-linear equation using bisection method correct upto three significant figure x2 – e-x = 3.