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

Table 1

The examples considered in this study.

Test functions	Zeros
𝑓1(𝑥)=(sin𝑥)^2+𝑥	𝛼1=0
𝑓2(𝑥)=(1+𝑥)+cos𝜋𝑥2−√1−𝑥^2	𝛼2≈−0.728584046444826⋯
𝑓3(𝑥)=(sin𝑥)^2−𝑥^2+1	𝛼3≈1.404491648215341⋯
𝑓4(𝑥)=𝑒^−𝑥+sin(𝑥)−2	𝛼4≈−1.0541271240912128⋯
𝑓5(𝑥)=𝑥𝑒^−𝑥−0.1	𝛼5≈0.111832559158963⋯
𝑓6(𝑥)=√𝑥^4𝜋+8sin𝑥^2+𝑥+2𝑥^34−√+186+17	𝛼6=−2
𝑓7(𝑥)=√𝑥^2+2𝑥+5−2sin(𝑥)−𝑥^2+3	𝛼7≈2.331967655883964⋯
𝑓8(𝑥)=sin^−1𝑥^2−𝑥−12+1	𝛼8≈0.594810968398369⋯
𝑓9(𝑥)=(𝑥)−√sin22(𝑥+1)	𝛼9≈0.785398163397448⋯
𝑓10(𝑥)=𝑥^5+𝑥^4+4𝑥^2−15	𝛼10≈1.347428098968304⋯
𝑓11(𝑥)=𝑥−sin(cos(𝑥))+1	𝛼11≈−0.1660390510510295⋯
𝑓12(𝑥)=𝑥^5−10costan^−1(𝑥)	𝛼12≈1.419315939421532⋯
𝑓13(𝑥)=𝑥^5−10costan^−1𝑥^4+17𝑥	𝛼13≈0.580593457809920⋯
𝑓14(𝑥)=sincostan^−12𝑥^sin(𝑥)−2𝑥	𝛼14≈0.273402731005321⋯
𝑓15(𝑥)=𝑥^3−𝑥^2−2𝑥−cos(𝑥)+2	𝛼15≈0.498542523582153⋯
𝑓16(𝑥)=√𝑥^3+sin(𝑥)−30	𝛼16≈9.716501993365200⋯
𝑓17(𝑥)=tan^−1𝑥^2−𝑥	𝛼17=1
𝑓18(𝑥)=sin^−1𝑥^2−2𝑥	𝛼18=0
𝑓19(𝑥)=sin−3𝑥cos√𝑥	𝛼19≈−0.74886062246711⋯
𝑓20(𝑥)=𝑥^2−cos(𝑥)	𝛼20≈0.89055270972295⋯