Numerical Method TU Board 2081 question paper

What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve

$$x^2–5x+6=0$$. Calculate upto 3 decimal places.

What are the limitations of direct methods for solving a system of linear equations? How Gauss Seidel method differs from Jacobi iteration? Solve the following system of linear equation using Jacobi iteration method.

2x-7y-10z=-17

5x+y+3z=14

x+10y+9z=7

Write an algorithm and program to implement Lagrange interpolation method.

Consider the following data points estimate the f(0.6) using Newton's interpolation formula.

x
0.1
0.2
0.3
0.4
0.5
f(x)
2.68
3.04
3.38
3.69
3.97

What is regression analysis? Fit a second order polynomial for the following data values.

x
2
4
6
8
10
y
1.4
2.0
2.4
2.6
2.8

What is numerical differentiation? The table below gives the values of distance travelled by a vehicle at various time interval, estimate the velocity and acceleration at x = 4.

Time(x)
1
2
4
8
10
Distance(y)
0
1
5
21
27

What is an application of numerical integration? Find the value of the integral
$${\int }_1^2\frac{e^x}{x}dx$$
using Simpson's $\frac{3}{8}$ rule with $n=6$.

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

x+2y-3z=4

2x+4y-6z=8

x-2y+5z=4

Given the data points below

X
1.0
3.0
4.0
f(x)
1.5
4.5
9.0

Find cubic spline which belongs to 1<=x<=3 and estimate f(2) using cubic splines.

What is differential equation? Differentiate between ODE and PDE with example.

Solve $\frac{dy}{dx}=\frac{x}{y},y(0)=1$, at $x=0.4$ using Runge–Kutta's $4^{\text{th}}$ order method.

Solve the Poisson equation
$$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=-64xy,0\le x\le 1,0\le y\le 1$$
with boundary conditions:
$$u(0,y)=0,u(x,0)=0,u(1,y)=150,u(x,1)=150\text{and}h=\frac{1}{3}.$$