Cubic convergence of parameter-controlled Newton-secant method for multiple zeros. (English) Zbl 1194.65073

Authors’ abstract: Let \(\mathbb C\to\mathbb C\) have a multiple zero a with integer multiplicity \(m\geq 1\) and be analytic in a sufficiently small neighborhood of \(\alpha\). For parameter-controlled Newton-secant method defined by
\[ x_{n+1}=x_n-\frac{\lambda f(x_n)^2}{f'(x_n)\cdot \{f(x_n)-f(x_n-\mu f(x_n)/f'(x_n))\}'}\;,\quad n= 0,1,2,\dots \]
we investigate the maximal order of convergence and the theoretical asymptotic error constant by seeking the relationship between parameters \(\lambda\) and \(\mu\). For various test functions, the numerical method has shown a satisfactory result with high-precision mathematica programming.


65H05 Numerical computation of solutions to single equations
65H04 Numerical computation of roots of polynomial equations


Full Text: DOI


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