(Solved): Numerical Analysis 4. Consider the fixed point iteration methods for solving f(x)=x3x2+x1=0, ...

4. Consider the fixed point iteration methods for solving $f(x)=x^3?x^2+x?1=0,$ starting from the initial guess $x_0=2$ and using - $g(x)=x?f(x)$ - $g(x)=x?f(x)/10$ - $g(x)=x?f(x)/2$ - $g(x)=x?f(x)/f^{?}(x)$ (a) Implement the iterations in MATLAB (sample code posted on Canvas). For each $g(x)$, output the errors, convergence rate, and converegence order at each iteration step. (b) Based on your numerical results from (a), for which $g(x)$ option(s), the iteration is convergent? Which $g(x)$ converges faster than the others? (c) For each $g(x)$, use the fixed-point iteration theorem to determine the convergence of iterations. What is the expected convergence rate? Does this match your numerical results above? clearvars $\frac{?}{0}$ removes all variables from the current $\mathrm{w}$ format long o more digits after the decimal point fo clc $\frac{?}{?}$ clean command window $f=@(x)x^{?}3?x^{?}2?1;$ $g=@(x)?x^{?}3+x^{?}2+1;$ $f\mathrm{p}=\mathrm{a}(\mathrm{x})\left(3?{\mathrm{x}}^{?}2\right)?2?\mathrm{x}+1;$ o derivative of $\mathrm{f}(\mathrm{x})$ $gp=a(x)?x(3?x?2);\frac{o}{0}$ derivative of $g(x)$ $\mathrm{x}$ _exact $=$ fzero(f, 1$)$; $\frac{?}{0}$ exact solution ? initialize the simulation $\mathrm{x}0=2$; $\mathrm{Nmax}=10$; $\mathrm{x}=\mathrm{NaN}(\mathrm{Nmax},1)$; $\mathrm{x}(1)=\mathrm{x}0$; ? fixed point iterations for $k=2:$ Nmax $\mathrm{x}(\mathrm{k})=\mathrm{g}(\mathrm{x}(\mathrm{k}?1))$; end ? calculate the absolute error and order below ???? ? output to command window ? $\mathrm{x}$