(Solved): Algorithm 1: Design the Newton's algorithm to determine the roots of the following non-linear equat ...

Algorithm 1: Design the Newton's algorithm to determine the roots of the following non-linear equation $(x?2{)}^2+4x?8=0$ when $\mathrm{x}$ is in the range [0,4]. The output of the algorithm should be a table that presents the following values $k,x_k,f\left(x_k\right),f^{?}\left(x_k\right)$ and $h_k$. For your algorithm, use a convergence tolerance of 1e-5. Algorithm 2: Design the Newton's method to determine the solution for the following system of equations $\begin{array}{l}{x}_1+2x_2?2=0\\ x_1^2+4x_2^2?4=0\end{array}$ Use a tolerance of $1\mathrm{e}?4$ in your algorithm. The algorithm should print the $k,\left[\begin{array}{ll}{x}_1^k& x_2^k\end{array}\right]$ in each iteration in a tabular format. Note: 1. The algorithms must work for a general equation(s). 2. You cannot use a built-in function to calculate either of the algorithms. 3. The program should print the tables in the two algorithms. To pass this assessment, you must: 1. Include the complete code for the algorithms. 2. The functioning of the algorithm is illustrated by applying it to the above problems. 3. Obtain the correct final solution.