Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations

Table 1

Test functions and their roots.

Function	Root
𝑓1(𝑥)=(𝑥−2)(𝑥^10+𝑥+1)exp(−𝑥−1),	𝛼=2
𝑓2(𝑥)=exp(−𝑥^2+𝑥+2)−cos(𝑥+1)+𝑥^3+1,	𝛼=−1
𝑓3(𝑥)=sin(𝑥)^2−𝑥^2+1,	𝛼=1.40449165⋯
𝑓4(𝑥)=𝑒^−𝑥−cos(𝑥),	𝛼=−0.666273126⋯
𝑓5(𝑥)=ln(𝑥^2+𝑥+2)−𝑥+1,	𝛼=4.15259074⋯
𝑓6(𝑥)=exp(𝑥^2+7𝑥−30)−1,	𝛼=1.40449165⋯
𝑓7(𝑥)=cos(𝑥)^2−𝑥5	𝛼=1.08598268⋯
𝑓8(𝑥)=sin(𝑥)−𝑥2,	𝛼=0
𝑓9(𝑥)=𝑥^10−𝑥^3−𝑥+1	𝛼=0.591448093⋯
𝑓10(𝑥)=sin(𝑥)−𝑥+1	𝛼=2.63066415⋯