Cubic convergence of parameter-controlled Newton-secant method for multiple zeros

Let f:C→Cf:C→C have a multiple zero αα with integer multiplicity m≥1m≥1 and be analytic in a sufficiently small neighborhood of αα. For parameter-controlled Newton-secant method defined by xn+1=xn−λf(xn)2f′(xn)⋅{f(xn)−f(xn−μf(xn)/f′(xn))},n=0,1,2,…, we investigate the maximal order of convergence and the theoretical asymptotic error constant by seeking the relationship between parameters λλ and μμ. For various test functions, the numerical method has shown a satisfactory result with high-precision Mathematica programming.