### Exam Year

Tribhuvan University

Institute of Science and Technology

2079

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 secant method can approximate the root of a non-linear equation? Explain with necessary derivation. Estimate a real root of following equation using secant method. Assume error precisionn of 0.01.

x3 + 2x – cos(x) = 4

2

How spline interpolation differs with the Langrage’s interpolation? Estimate the value of f(0) and f(4) using cubic spline interpolation from the following data.

 x -1 1 2 3 f(x) -10 -2 14 86
3

What is pivoting? Why is it necessary? Write an algorithm and program to solve the set of n linear equations using Gaussian elimination method.

Section B

Attempt any eight questions.

4

Calculate a real root of the following function using bisection method correct upto 3 significant figures.

x2 – e-x = 3

5

What is fixed point iteration method? How can it converge to the root of a non-linear equation? Also explain the diverging cases with suitable examples.

6

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

7

Fit the quadratic function for the data given below using least square method.

 x 1 1.5 2 2.5 3 3.5 4 f(x) 2.7 4 5.8 8.3 11.2 15 19
8

Estimate the integral value of following function from x =1.2 to 2.4 using Simpson’s 1/3 rule.

 x 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 f(x) 1.53 2.25 3.18 4.32 5.67 7.23 8.98 10.94 13.08
9

What is Gaussian integration formula? Evaluate the following integration using Gaussian integration three ordinate formula

01 Sinx/x dx

10

Solve the following set of equations using Gauss Siedal method.

x + 2y + 3z = 4

6x + 4y + 5z = 16

5x + 2y + 3z = 12

11

Solve the following differential equation for 0 ≤ x ≤ 1 taking h=0.5 using Runge Kutta 4th order method.

y'(x) + y = 3x with y(0)=2

12

Solve the Poisson’s equation ∇2f=3x2y over the square domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f=0 on the boundary and h=1.