Explain how Newton-Raphson method differs from bisection method. Derive the formula for Newton-Raphson method. Use bisection method to solve \(x^3 - x - 2 = 0\) in \(1< x < 2\).
Write an algorithm and program to compute interpolation using Newton's backward interpolation.
Explain the power method for finding eigen value. Differentiate between Jacobi and Gauss-Seidel method. Solve the following given system of linear equation using Jacobi method.
\[5x + y + z = 7\] \[x + 4y + z = 6\] \[x + y + 3z = 5\]
Attempt any Eight questions
[8×5=40]Differentiate between absolute and relative error. Explain fixed point iteration method.
Fit a second-degree polynomial using least square method for: \((-1,2),\ (0,1),\ (1,2),\ (2,5),\ (3,10)\).
Find first and second derivative at \(x = 2\) from the following tabulated data value:
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 |
| \(f(x)\) | 1 | 3.375 | 8 | 15.625 | 27 |
What is numerical integration? Derive Newton-Cote's quadrature formula.
Integrate \(\displaystyle\int_{1}^{2} \frac{e^x}{x}\, dx\) using the trapezoidal rule with \(n=5\).
Solve following system of linear equation by Gauss elimination with partial pivoting.
\[2x + 3y + z = 9\] \[x + 4y + 5z = 15\] \[3x + y + 2z = 10\]
Use the differential equation \(\dfrac{dy}{dx} = 2x + 3y\) with initial condition \(y(0) = 1\). Find \(y(0.2)\) using the Runge-Kutta method.
Differentiate between initial value and boundary value problem. Explain solution of system of ODEs.
Explain the classification of partial differential equations by order and linearity.