What is non-linear equation? Derive the required expression to calculate the root of non-linear equation using secant method. Using this expression find a root of following equation.

\(x^{2} + cos(x) – e^{-x} – 2 = 0\)

What is matrix factorization? Factorize the given matrix A into LU using Doolittle algorithm and also solve Ax = b for given b using L and U matrices.

A = \(\begin{bmatrix}2 & 4 & -4 & 0\\ 1 & 5 & -5 & -3\\ 2 & 3 & 1 & 3\\ 1 & 4 & -2 & 2\end{bmatrix}\) and b = \(\begin{bmatrix}12\\ 18\\ 8\\ 8\end{bmatrix}\)

What is initial value problem and boundary value problem? Write an algorithm and program to solve the boundary value problem using shooting method.

Calculate a real negative root of following equation using Newton’s method for polynomial.

\(x^4 + 2x^3 + 3x^2 + 4x = 5\)

What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.

Find the lowest degree polynomial, which passes through the following points:

Using this polynomial estimate f(x) at x = 0

The fit function of type y = a + bx for the following points using the least square method.

Calculate the integral value of the function given below from x = 1.8 to x = 3.4 using Simpson’s 1/3 rule.

Evaluate the following integration using Romberg integration.

\(\int_{0}^{1} \frac{sinx}{x} dx\)

Solve the following set of equations using Gauss Seidel method.

x + 2y + 3z = 4

6x – 4y + 5z = 10

5x + 2y + 2z = 25

From the following differential equation estimate y(1) using RK 4^th order method.

\(\frac{dy}{dx} + 2x^2y = 4 \enspace with \enspace y(0) = 1\)

[Take h = 0.5]

Solve the Poison’s equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.