Optimized Steffensen-Type Methods with Eighth-Order Convergence and High Efficiency Index

Table 1

The examples considered in this study.

Test functions	Simple zeros
𝑓1(𝑥)=(sin𝑥)^2+𝑥	𝛼1=0
𝑓2(𝑥)=(1+𝑥)+cos(𝜋𝑥/2)−√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^4+8sin(𝜋/(𝑥^2+2))+(𝑥^3/(𝑥^4+1))−√6+(8/17)	𝛼6=−2
𝑓7(𝑥)=√𝑥^2+2𝑥+5−2sin(𝑥)−𝑥^2+3	𝛼7≈2.331967655883964⋯
𝑓8(𝑥)=sin^−1(𝑥^2−1)−𝑥/2+1	𝛼8≈0.594810968398369⋯
𝑓9(𝑥)=(sin(𝑥)−√2/2)(𝑥+1)	𝛼9≈0.785398163397448⋯
𝑓10(𝑥)=𝑥−sin(cos(𝑥))+1	𝛼10≈−0.1660390510510295⋯
𝑓11(𝑥)=𝑥^5−10cos(tan^−1(𝑥^4))+17𝑥	𝛼11≈0.580593457809920⋯
𝑓12(𝑥)=sin(cos(tan^−1(2𝑥^sin(𝑥))))−2𝑥	𝛼12≈0.273402731005321⋯
𝑓13(𝑥)=𝑥^3−𝑥^2−2𝑥−cos(𝑥)+2	𝛼13≈0.498542523582153⋯
𝑓14(𝑥)=√𝑥^3+sin(𝑥)−30	𝛼14≈9.716501993365200⋯
𝑓15(𝑥)=tan^−1(𝑥^2−𝑥)	𝛼15=1
𝑓16(𝑥)=sin^−1(𝑥^2)−2𝑥	𝛼16=0