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Variants of Newton’s method for functions of several variables. (English) Zbl 1123.65042

Newton-like methods are discussed for finding a real solution of a system of nonlinear equations: $F\left(x\right)=0$ in ${ℝ}^{n}$. The authors propose a midpoint Newton method:

${x}^{\left(k+1\right)}={x}^{\left(k\right)}-{J}_{F}{\left(\left({x}^{\left(k\right)}+{z}^{\left(k\right)}\right)/2\right)}^{-1}F\left({x}^{\left(k\right)},\phantom{\rule{1.em}{0ex}}k=0,1,\cdots ·$

Here ${J}_{F}\left(x\right)$ is the Jacobian matrix of the function $F$. ${z}^{\left(k\right)}$ is defined via a Newton step as

${z}^{\left(k\right)}={x}^{\left(k\right)}-{J}_{F}{\left({x}^{\left(k\right)}\right)}^{-1}F\left({x}^{\left(k\right)}\right)·$

The midpoint Newton method is proven to be of quadratic covergence and illustrated to be better than the Newton method itself with numerical examples. But it is not compared with the two step Newton method and the cost of evaluation of the function $F$ and its Jacobian at each iteration step is not considered.

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