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

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