Note: All of the questions are base on coding and The termination tolerance is ${10}^{-10}$ for all tests, and the $n$ represents $n^{th}$ iteration.

Code for LU decomposition to solve $nn$ linear system. $($You can use the data from last test to check correctness$).$

Given the nonlinear equation $5x+xsin(x)+12=0,$ code for Newton, Secant, Picard, and one other fixed$-$point method. Note: Your code should output the solution and the convergence plot $(n$ vs $log(|x_{n+1}-x_n|)),$ and your plot should be a PDF file. The initial guess for Newton is $x_0=2,$ you need to propose the initial guess for the Picard and other fixed$-$point method by yourself.

Given a nonlinear system:

$x{e}^{-x+y}-y=1$

$x-y^2=4$

code for Newton, Picard, and one other fixed$-$point method. Note: Your code should output the solution and the convergence plot $(n$ vs $log(max\{|x_{n+1}-x_n|,|y_{n+1}-y_n|\})),$ and your plot should be a PDF file. The initial guess for Newton and Picard is vec$(x_0)=([1\mathrm{.}5],[1]),$ you need to propose the initial guess for the other fixed$-$point method by yourself.