Numerical Method TU Board 2080 question paper

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. $y^{\prime \prime }+x{y}^{\prime }-xy=2x$, 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. $${\int }_2^{2.6}{\int }_4^{4.4}\frac{dxdy}{xy}$$

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: X12345F(x)26122030

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. X13579y-1.2012.80119.60472.801302.80

Find the Eigen values and Eigen vectors of the Matrix: $$A=\left[\begin{array}{cc}3& -1\\ 1& 1\end{array}\right]$$

Solve the Poisson's equation $\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}=2x^2{y}^2$ over the square domain $0\le x\le 3$ and $0\le y\le 3$ with $f=0$ on the boundary and $h=1$.