Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2079

Bachelor Level / Second Year / Third Semester / Science

B.Sc in Computer Science and Information Technology (CSC212)

(Numerical Method)

Full Marks: 60

Pass Marks: 24

Time: 3 Hours

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

The figures in the margin indicate full marks.

Section A

Long Answers Questions

Attempt any TWO questions.
[2*10=20]
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 precision of 0.01.

$x^3 + 2x - \cos(x) = 4$
[10]
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.

$\begin{array}{c|cccc} x & -1 & 1 & 2 & 3 \\ \hline f(x) & -10 & -2 & 14 & 86 \end{array}$
[10]
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.[10]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Calculate a real root of the following function using bisection method correct upto 3 significant figures.

$x^2 - e^x = 3$
[5]
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. [5]
6.
Write down the program for solving ordinary differential equation using Heun's method. [5]
7.
Fit the quadratic function for the data given below using least square method.

$\begin{array}{c|ccccccc} x & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 \\ \hline f(x) & 2.7 & 4.0 & 5.8 & 8.3 & 11.2 & 15.0 & 19.0 \end{array}$
[5]
8.
Estimate the integral value of following function from x = 1.2 to 2.4 using Simpson's 1/3 rule.

$\begin{array}{c|cccccccc} x & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 & 2.6 \\ \hline f(x) & 1.53 & 2.25 & 3.18 & 4.32 & 5.67 & 7.23 & 8.98 & 10.94 & 13.08 \end{array}$
[5]
9.
What is Gaussian integration formula? Evaluate the following integration using Gaussian integration three ordinate formula.

$\int_{0}^{1} \frac{\sin x}{x} dx$
[5]
10.
Solve the following set of equations using Gauss Siedal method.

$x + 2y + 3z = 4$

$6x + 4y + 5z = 16$

$5x + 2y + 3z = 12$
[5]
11.
Solve the following differential equation for $0 \leq x \leq 1$ taking $h=0.5$ using Runge Kutta 4th order method.

$y'(x) + y = 3x \text{ with } y(0) = 2$
[5]
12.
Solve the Poisson's equation $\nabla^2 f = 3x^2y$ over the square domain $0 \leq x \leq 3$, $0 \leq y \leq 3$ with $f=0$ on the boundary and $h=1$. [5]