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

Geum, Young Hee; Kim, Young Ik

J. Comput. Appl. Math. 233, No. 4, 931-937 (2009).

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.

Reviewer: B. Döring (Düsseldorf)

MSC:

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

Keywords:

parameter-controlled; leap-frogging Newton’s method; multiple zero; numerical examples; secant method; convergence; asymptotic error

Software:

Mathematica

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