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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

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