An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

Table 1

Test functions.

𝑓(𝑥)	𝑟
𝑓1(𝑥)=𝑥^5+𝑥^4+4𝑥^2−15	1.3474280989683050
𝑓2(𝑥)=sin(𝑥)−𝑥/3	2.2788626600758283
𝑓3(𝑥)=10𝑥𝑒^−𝑥2−1	1.6796306104284499
𝑓4(𝑥)=cos(𝑥)−𝑥	0.7390851332151606
𝑓5(𝑥)=𝑒^−𝑥2+𝑥+2−1	−1.0000000000000000
𝑓6(𝑥)=𝑒^−𝑥+cos(𝑥)	1.7461395304080124
𝑓7(𝑥)=ln(𝑥^2+𝑥+2)−𝑥+1	4.1525907367571583
𝑓8(𝑥)=arcsin(𝑥^2−1)−𝑥/2+1	0.5948109683983692
𝑓9(𝑥)=𝑥𝑒^𝑥2−sin^2𝑥+3cos𝑥+5	−1.2076478271309189