Questions
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Qns. By Syllabus

Semester

Subject

Year

Bachelors Level/Second Year/Third Semester/Science csit/third semester/numerical method/syllabus wise questions

B.Sc Computer Science and Information Technology

Institute of Science and Technology, TU

Numerical Method (CSC212)

Year Asked: 2081, syllabus wise question

Interpolation and Regression
1.
Write an algorithm and program to implement Lagrange interpolation method. [10]
2.
Consider the following data points. Estimate $f(0.6)$ using Newton's interpolation formula.

$\begin{array}{c|ccccc} x & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\ \hline f(x) & 2.68 & 3.04 & 3.38 & 3.69 & 3.97 \end{array}$
[5]
3.
What is regression analysis? Fit a second order polynomial for the following data values.

$\begin{array}{c|ccccc} x & 2 & 4 & 6 & 8 & 10 \\ \hline y & 1.4 & 2.0 & 2.4 & 2.6 & 2.8 \end{array}$
[5]
4.
Given the data points below, find cubic spline which belongs to $1 \leq x \leq 3$ and estimate $f(2)$ using cubic splines.

$\begin{array}{c|ccc} x & 1.0 & 3.0 & 4.0 \\ \hline f(x) & 1.5 & 4.5 & 9.0 \end{array}$
[5]
Numerical Differentiation and Integration
1.
What is numerical differentiation? The table below gives the values of distance travelled by a vehicle at various time intervals. Estimate the velocity and acceleration at $x = 4$.

$\begin{array}{c|ccccc} \text{Time}(x) & 1 & 2 & 4 & 8 & 10 \\ \hline \text{Distance}(y) & 0 & 1 & 5 & 21 & 27 \end{array}$
[5]
2.
What is an application of numerical integration? Find the value of the integral using Simpson's 3/8 rule with $n = 6$.

$\int_{1}^{2} \frac{e^x}{x} dx$
[5]
Solution of Nonlinear Equations
1.
What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve the following equation up to 3 decimal places:

$x^2 - 5x + 6 = 0$
[10]
Solution of Ordinary Differential Equations
1.
What is differential equation? Differentiate between ODE and PDE with example. [5]
2.
Solve $\frac{dy}{dx} = \frac{x}{y}$, $y(0) = 1$, at $x = 0.4$ using Runge-Kutta's 4th order method.

$\frac{dy}{dx} = \frac{x}{y}, \quad y(0) = 1, \quad x = 0.4$
[5]
3.
Solve the Poisson equation with boundary conditions:

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -64xy, \quad 0 \leq x \leq 1, \; 0 \leq y \leq 1$

$u(0,y) = 0, \; u(x,0) = 0, \; u(1,y) = 150, \; u(x,1) = 150, \; h = \frac{1}{3}$
[5]
Solving System of Linear Equations
1.
What are the limitations of direct methods for solving a system of linear equations? How does Gauss Seidel method differ from Jacobi iteration? Solve the following system of linear equations using Jacobi iteration method:

$2x - 7y - 10z = -17$

$5x + y + 3z = 14$

$x + 10y + 9z = 7$
[10]
2.
Solve the following system of linear equations using Gauss-Jordan elimination method:

$x + 2y - 3z = 4$

$2x + 4y - 6z = 8$

$x - 2y + 5z = 4$
[5]