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On Generalizations of the Newton-Raphson-Simpson Method by DeFranco, Mario is a scholarly article available to read on EtoBox.

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We present generalizations of the Newton-Raphson-Simpson method. Specifically, for a positive integer $m$ and the sequence of coefficients of a Taylor series of a function $f(z)$, we define an algorithm we denote by NRS($m$) which is a way to evaluate, in our terminology, a sum of $m$ formal zeros of $f(z)$. We prove that NRS(1) yields the familiar iterations of the Newton-Raphson-Simpson method. We also prove that NRS($m$) is way to evaluate certain $\mathscr{A}$-hypergeometric series defined by Sturmfels. In order to define these algorithms, we make use of combinatorial objects which we call trees with negative vertex degree.

Author

DeFranco, Mario

Published

2019

Language

EN