Fit a second order polynomial to the data in the table below: [5]

Solve the double integration using Simpson's 1/3 rule. $\int_{2}^{2.6} \int_{4}^{4.4} \frac{dxdy}{xy}$ [5]

Why Numerical Integration is required? Compute the integral: $I = \int_{-1}^{1} e^x dx$ using composite trapezoidal rule for n = 4. [5]

Evaluate $\frac{dy}{dx}$ at x = 5 using Newton's forward interpolation formula using the following table. [5]

Write an algorithm and a C-Program to obtain roots of non-linear equation using Newton Raphson Method. [10]

Use secant method to estimate the root of the equation $x^2-5x+6=0$, with initial estimate $x_1 = 4$ and $x_2 = 2$ (EPS=0.05). [5]

What are the sources of errors? Discuss various types of errors encounters in numerical computation. [5]

Solve the following ordinary differential equation using shooting method. $y'' + xy' - xy = 2x$ with boundary conditions $y(0)=1$ and $y(2)=10$ [10]

Solve the following differential equation $ \frac{dy}{dx} = 3x + \frac{y}{2} $ with $ y(0) = 1 $ for $ x = 0.2 \quad (h = 0.1) $ using Euler's Method. [5]

Solve the Poisson's equation $\partial^2f/\partial x^2+\partial^2f/\partial y^2 = 2x^2y^2$ over the square domain 0<=x<=3 and 0<=y<=3 with f=0 on the boundary and h = 1. [5]

Compare and contrast between Jacobi iterative methods and Gauss Seidal method? Solve the following equation using Gauss Seidal method. $x+2y+3z = 5, 2x+8y+22z = 6$ and $3x+22y+82z = -10$ [10]

Find the Eigen values and Eigen vectors of the Matrix: $A=\begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}$ [5]