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 \(3x + 22y + 82z = -10\)
Attempt any Eight questions
[8×5=40]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.
\[\int_{2}^{2.6} \int_{4}^{4.4} \frac{dx\, dy}{xy}\]
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:
| X | 1 | 2 | 3 | 4 | 5 |
| F(x) | 2 | 6 | 12 | 20 | 30 |
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.
| X | 1 | 3 | 5 | 7 | 9 |
| y | -1.20 | 12.80 | 119.60 | 472.80 | 1302.80 |
Find the Eigen values and Eigen vectors of the Matrix:
\[A = \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}\]
Solve the Poisson's equation \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 2x^2y^2\) over the square domain \(0 \leq x \leq 3\) and \(0 \leq y \leq 3\) with \(f=0\) on the boundary and \(h = 1\).