Question $2$

$($a$)$ Write a function that implements the following pseudo$-$code:

Input: $f,f^{\text{'}},x,lon,N.$

Output: $x^{**}.$

Repeat $N$ times:

$($a$)$ Set $y_1=x.$

$($b$)$ Take one Newton step, starting from $y_1.$ Call the result $y_2.$

$($c$)$ Take one Newton step, starting from $y_2.$ Call the result $y_3.$

$($d$)$ Set

$x=y_1-\frac{(y_2-y_1{)}^2}{y_3-2y_2+y_1}$

$($e$)$ Display $|f(x)|.$

$($f$)$ If $|f(x)|<mi\; id="EditorElement\_2588295"></mi><mi\; id="EditorElement\_2588297">l</mi><mi\; id="EditorElement\_2588299">o</mi><mi\; id="EditorElement\_2588301">n</mi>$ print "converged!", break.

Output $x^{**}=x.$

This algorithm is called Steffensen's iteration.

$($b$)$ Test your routine on the problem

exp$(-x^2+x)-\frac{1}{2}x=1\mathrm{.}0836(with$ initial guess $x=1)$

Show that Newton iteration does not converge quadratically, but your new iterative algorithm

does.