Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods

Algorithm 1

f[x_]:=Sin[x]Cot[x]+Cos[x^∧2]+0.5;

a=-2.; b=12.;

zeros=Reap[soln=y[x]/.First[NDSolve[{y ′ [x]==Evaluate[D[f[x],x]],y[b]==

(f[b])},y[x],{x,a,b},

   Method->{"EventLocator","Event"->y[x],

"EventAction":>Sow[{x,y[x]}]}]]][[2,1]];

initialPoints=Sort[Flatten[Take[zeros,Length[zeros],1]]]