Problem Set - Picard and Newton iteration

Problem Set - Picard and Newton iteration

 Let $f:\left(0,\frac{1}{2}\pi \right)\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\mathrm{tan}x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in \left(0,\frac{1}{2}\pi \right)$ using Picard iteration. Let $f:\left(0,\frac{1}{2}\pi \right)\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\mathrm{tan}x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in \left(0,\frac{1}{2}\pi \right)$ using Newton iteration (let $F\left(x\right)=x-f\left(x\right)$). Show that the equation $g\left(x\right)={x}^{3}+x-1=0$ has a solution between 0 and 1. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:\left[0,1\right]\to \left[0,1\right]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=1/\left({x}^{2}+1\right)$). Show that the equation $g\left(x\right)={x}^{4}-4{x}^{2}-x+4=0$ has a solution between $\sqrt{3}$ and 2. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:\left[\sqrt{3},2\right]\to \left[\sqrt{3},2\right]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=\sqrt{2+\sqrt{x}}$).