Problem $3(30$ point$)$

Consider

$f(x)=x^2+6x+8$

$($i$)10$ point Construct the MATLAB code of the Newton$-$Rhapson method to find the root of $f.$ Discuss the convergence of the Newton$-$Rhapson algorithm.

$($ii$)10$ point Construct the MATLAB code of the secant method to find the root of $f.$

$($iii$)10$ point Iterative $($i$)-($ii$)$ for two times when the initial conditions are $p_0=-3\mathrm{.}8$ and $p_0=-1\mathrm{.}8.$ Compare the two algorithms in terms of the absolute error.

Solution

$($i$)$ MATLAB code

Convergence: $f$ is twice differentiable. Also, $f^{\text{'}}(p)0,$ where $p$ is the root of $p.$ Hence, by the Newton$-$Rhapson theorem, there exists a $>0$ such that the algorithm converges to the root of $f$ for any initial condition of $p_{0}in[p-,p+].$

$($ii$)$ MATLAB code

$($iii$)$ simple computation