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

6x_{1} – 2x_{2} + x_{3} = 11

x_{1} + 2x_{2} – 5x_{3} = -1

-2x_{1} + 7x_{2} + 2x_{3} = 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^{2 }+ 1 ; y(0) = 2, when

- h = 0.5
- 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 | 0.6990 |

y | 1.4440 | 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 | 1.0 | 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 x^{2} – e^{-x }= 3.

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