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An analysis of the properties of the variants of Newton’s method with third order convergence. (English) Zbl 1123.65036

This is a nice summary of the third order iterative methods for solving nonsingular nonlinear equations $f\left(x\right)=0$. The authors formulate eight third order methods into a single frame ${x}_{k+1}={x}_{k}-f\left({x}_{k}\right)/{D}_{m}\left({x}_{k}\right)$ with eight different denominators ${D}_{m}\left(x\right)$, $m=1,2,\cdots ,8$, approximating the Jacobian of $f$ from different perspectives. In addition to convergence rates, the information usage and efficiency, i.e., number of function and Jacobian values required per iteration and comparison among the convergence orders, are evaluated.

It is shown that these third order methods are variations of the Halley method and are all contractive in the same neighbourhood. The extension of these methods to systems of equations is also discussed. Convergence rates and neighbourhoods are illustrated from numerical and geometrical perspectives by various examples.

MSC:
 65H05 Single nonlinear equations (numerical methods) 65H10 Systems of nonlinear equations (numerical methods)