Numerical Method TU Board 2021 question paper

Compute the root of equation $x^2-5x+6=0$ in the vicinity of $x=5$ using Newton-Raphson method.

Estimate the value of $\ln (2.5)$ using Newton-Gregory forward difference formula given the following data:X1.02.03.04.0Ln(x)0.00.69311.09861.3863

Write an algorithm to compute integration using Simpson's 1/3 rule. Evaluate integration of ${\int }_0^3(3x^2+2x-5)dx$ using Simpson's 1/3 rule.

What is meant by ill-conditioned system? Find the Cholesky decomposition of the following matrix: $$\left[\begin{array}{ccc}4& 1& 1\\ 1& 5& 2\\ 1& 2& 3\end{array}\right]$$

Use the classical RK method to estimate $y(0.4)$ of the equation $\frac{dy}{dx}=x^2+y^2$ with $y(0)=0$, assume $h=0.2$.

Solve the Poisson equation ${\nabla }^{2}f=2x^2{y}^2$ over the square domain $0\le x\le 3and0\le y\le 3$ with $f=0$ on the boundary and $h=1$.

Write a short note on (Any Two): a) Types of Errors b) Convergent Methods c) PDE

Write an algorithm and program to compute root of nonlinear equation using bisection method.

Given the data points:i012x_i91625f_i345Estimate the function value of $f$ at $x=14$ using cubic splines.

a) Solve the following system of equations using Gauss-Seidel method: 2x - 7y - 10z = -17 5x + y + 3z = 14 x + 10y + 9z = 7 b) Write an expansion of Taylor's Theorem. Solve the equation: $y^{\prime }(x)=x^2+y^2,y(0)=1$ for $x=0.5$ using Taylor method.