Empty Regions of Polynomial Zeros with Derivatives and the Newton Method

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  • Yamamoto Osami
    Department of Information System Engineering, Faculty of Engineering, Aomori University

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  • 多項式の微分係数情報を付加した零点不在領域とNewton法
  • タコウシキ ノ ビブン ケイスウ ジョウホウ オ フカ シタ レイテン フザイ リョウイキ ト Newtonホウ

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Abstract

The empty regions used in this paper are simple extensions of the empty regions that are used in [5]. The region around a real number α depends on f(α), f′(α) and ‖f‖, where ‖f‖ denotes the polynomial norm of f. We show that for almost all polynomials of degree greater than 1 with real coefficients, the Newton method is locally equivalent to a simple algorithm, which uses the empty regions. Moreover, analyzing the region, we obtained a modified Newton's formula such that

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