A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

Table 1

Various methods with typical choice of parameters (𝜆,𝜌,𝑎,𝑏,and𝑐).

Method	Parameter (𝜆,𝜌)	(𝑎,𝑏,𝑐)
		𝑚𝑎=1𝑚+2𝑚+8𝑚(𝑚+2)^3(5𝜅−1)−(𝑚+2)^2𝜙16𝑚𝜅^1,
Y1	(1,−5)	𝜙1=𝑚−5(𝑚+2)𝜅)^2(10𝜅+𝑚(7𝜅−1),
		𝑏=−(𝑚+2)(𝑚−5(𝑚+2)𝜅)^316𝜅,; 1𝑐=8𝑚(𝑚+2)^3𝜅(1−5𝜅)
		𝑚𝑎=1𝑚+2𝑚+8𝑚(𝑚+2)^3(3𝜅−1)−(𝑚+2)^2𝜙16𝑚𝜅^2,
Y2	(1,−3)	𝜙2=𝑚−3(𝑚+2)𝜅)^2(6𝜅+𝑚(5𝜅−1),
		𝑏=−(𝑚+2)(𝑚−3(𝑚+2)𝜅)^316𝜅, 1𝑐=8𝑚(𝑚+2)^3𝜅(1−3𝜅)
		𝑚𝑎=1𝑚+2𝑚−8𝑚(𝑚+2)^3(10𝜅+1)−(𝑚+2)^2𝜙16𝑚𝜅^3,
Y3	(1,10)	𝜙3=(𝑚+20𝜅+8𝑚𝜅)(𝑚+10(𝑚+2)𝜅)^2,
		𝑏=−(𝑚+2)(𝑚+10(𝑚+2)𝜅)^316𝜅, 1𝑐=8𝑚(𝑚+2)^3𝜅(1+10𝜅)
Y4	(𝑚+2)^24(8+𝑚(5+𝑚))𝜅,0	𝑚0,−^3(8+𝑚(5+𝑚)),14(𝑚+2)8𝑚(𝑚+2)^3𝜅
Y5	−(𝑚+2)^2𝑚𝑚^2𝜅1+2𝑚+4,−𝜅	𝑚𝑚0,−^2+2𝑚+42(𝑚+2),0