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Modifications of Newton’s method to extend the convergence domain. (English) Zbl 1308.65072
Summary: The paper is devoted to description of certain ways of extending the domain of convergence of Newton’s method. This paper is a survey of contributions of representatives of Soviet and Russian mathematical school, namely, Kalitkin, Puzynin, Madorskij and others. They introduced different kinds of damping multiplier and showed that their usage may be helpful and beneficial while solving different nonlinear equations and systems starting with “bad” zero estimate. We have also paid attention to the problem of degeneracy of Jacobian matrix and the ways it was solved by named researchers. Finally, we have tested the presented iterative schemes on some examples in order to check their effectiveness. All complete strict proofs of key theorems can be found both in Russian and English in the provided bibliography.

MSC:
65H10 Numerical computation of solutions to systems of equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65H05 Numerical computation of solutions to single equations
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