## Polynomial interpolation in nondivision algebras

Gerhard Opfer

### Abstract

Algorithms for two types of interpolation polynomials in nondivision algebras are presented. One is based on the Vandermonde matrix, and the other is close to the Newton interpolation scheme. Examples are taken from $\mathbb{R}^4$-algebras. In the Vandermonde case, necessary and sufficient conditions for the existence of interpolation polynomials are given for commutative algebras. For noncommutative algebras, a conjecture is proposed. This conjecture is true for equidistant nodes. It is shown that the Newton form of the interpolation polynomial exists if and only if all node differences are invertible. Several numerical examples are presented.

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### Key words

interpolation polynomials in nondivision algebras, Vandermonde-type polynomials in nondivision algebras, Newton-type polynomials in nondivision algebras, numerical examples of interpolation polynomials in nondivision algebras of $\mathbb{R}^4$

### AMS subject classifications

15A66, 1604, 41A05, 65D05

### Links to the cited ETNA articles

 [4] Vol. 41 (2014), pp. 133-158 Drahoslava Janovská and Gerhard Opfer: Zeros and singular points for one-sided coquaternionic polynomials with an extension to other $R^4$ algebras [8] Vol. 36 (2009-2010), pp. 9-16 Gerhard Opfer: Polynomials and Vandermonde matrices over the field of quaternions

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