BIT Numerical Methods

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

Solve the following ordinary differential equation using shooting method.

\(y'' + xy' - xy = 2x\), with boundary conditions \(y(0)=1\); and \(y(2)=10\)

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 \(3x+22y+82z = -10\)

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).

Solve the double integration using Simpson's 1/3 rule. \(\int_{2}^{2.6} \int_{4}^{4.4} dxdy \,/\, xy\)

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

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

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

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

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

Find the Eigen values and Eigen vectors of the Matrix:

\(A = \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}\)

Solve the Poisson's equation \(d^2f/dx^2 + d^2f/dy^2 = 2x^2y^2\) over the square domain \(0 \le x \le 3\) and \(0 \le y \le 3\) with \(f=0\) on the boundary and \(h = 1\).