Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

Algorithm 1

The Mathematica code for finding the asymptotic error constant in Theorem 4.

Clear[“Global’^*”]

(*Assuming  e  =  x  -  α  and  𝚌_𝚓  =  (f^∧(j)  (α))/j!,  for  j = 1,  2,  …,  8*)

f[e_]∶=𝚌₁*e  +  𝚌₂*e^∧2  +  𝚌₃*e^∧3  +  𝚌₄*e^∧4  +  𝚌₅*e^∧5  +  𝚌₆*e^∧6  +  𝚌₇*e^∧7  +  𝚌₈*e^∧8;

(*Assuming  b  =  w  -  α*)

fe  =  f[e];  b  =  e  +  β  fe;  fw  =  f[b];  d  =  (fw - fe)/(β  fe);

(*Assuming  u = y - α*)

u = e-Series[fe/d,  {e,  0,  8}]//Simplify;

(*Assuming  v = z - α*)

fu = f[u];  v = e - ((fe²)/(d∗(fe - fu - (fu²)/fw)))//Simplify;

fv = f[v];  fyz =(fu - fv)/(u - v);

(*Assuming  e1=  Subscript[x,  new] - α*)

e1= v - (fv/(fv/(v - e)+ fyz))∗(1+(1/(1+ β  d))(fu/fe)^∧2

    +(2β  d +(β  d)^∧2)(fu/fw)^∧3+ fv/fw +𝛾(fv/fu)^∧2)//FullSimplify