Geodesic
$p$th-order approximation of the solution set of nonlinear equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 12, pp. 1951-1969

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Given a system of nonlinear equations, a formula is derived for the family of its approximate solutions of up to the pth order of smallness. A formula approximating an implicit function up to the third order of smallness is presented. Iterative methods converging with the $p$th order are constructed for solving systems of nonlinear equations. These results are extended to the degenerate case. Examples of applying the results are given. 
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     author = {Yu. G. Evtushenko and A. A. Tret'yakov},
     title = {$p$th-order approximation of the solution set of nonlinear equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1951--1969},
     publisher = {mathdoc},
     volume = {53},
     number = {12},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_12_a1/}
}
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Yu. G. Evtushenko; A. A. Tret'yakov. $p$th-order approximation of the solution set of nonlinear equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 12, pp. 1951-1969. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_12_a1/