### Exam Year

Tribhuvan University

Institute of Science and Technology

2078

Bachelor Level / third-semester / Science

Computer Science and Information Technology( CSC207 )

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 questions:

1

How can  Horner’s rule be used to evaluate the f(x) and f(x) of a polynomial at a given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner’s rule.

2

Write matrix factorization? How can be used to solve a system of linear equations? Factorize the given matrix A and solve the system of equations Ax = b for given b using L and U matrices.

A = $$\begin{bmatrix}1 & 2 & 3\\ 2 & 8 & 11\\ 3 & 22 & 36\end{bmatrix}$$ and b = $$\begin{bmatrix}4\\ 12 \\28\end{bmatrix}$$

3

What is a higher-order differential equation? How can you solve the higher-order differential equation? Explain. Solve the following differential equation for 1 ≤ x ≥ 2, taking h = 0.25

$$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 5y = 0$$, width y(1) = 1 and y(1) = 2

Section B

Attempt any EIGHT questions:

4

How the half-interval method can be estimate a root of a non-linear equation? Find a real root of the following equation using the half-interval method to correct up to two decimal places.

x2 – e-x – x = 1

5

Calculate the real root of the given equation using fixed point iteration correct up to 3 significant figures.

2x3 – 2x = 5

6

What is Newton’s interpolation? Obtain the divided difference table from the following data set and estimate the f(x) at x = 2 and x = 5.

 x 3.2 2.7 1 4.8 5.6 f(x) 22 17.8 14.2 38.3 51.7
7

What is linear regression? Fit the linear function to the following data

 x 1 1.2 1.4 1.6 1.8 2 2.2 2.4 f(x) 2 2.6 3.9 6 9.3 15 20.6 30.4
8

What are the problems with polynomial interpolation for a large number of data set? How such problems are addressed? Explain with an example.

9

Evaluate the following integration using Romberg integration.

$$\int_{0}^{1} \frac{sin^2 x}{x} dx$$

10

Solve the following set of linear equations using the Gauss-Jordan method.

x2 + 2x3 + 3x4 = 9

7x1 + 6x2 + 5x3 + 4x4 = 33

8x1 + 9x2 + x4 = 27

2x1 + 5x2 + 4x3 + 3x4 = 23

11

Solve the following differential equation for 1 ≤ x ≤ 2, taking h = 0.25 using Heun’s method.

y(x) + x2y = 3x, with y(1) = 1

12

Consider a metallic plate of size 90cm by 90cm. The two adjacent sides of the plate are maintained at a temperature of 1000C and the remaining two adjacent sides are held at 2000C. Calculate the steady-state temperature at interior points assuming a grid size of 30 cm by 30 cm.