MANIPULATING POLYNOMIALS IN GENERALIZED FORM

Gemignani, Luca (1996) MANIPULATING POLYNOMIALS IN GENERALIZED FORM. Technical Report del Dipartimento di Informatica . Università di Pisa, Pisa, IT.

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Abstract

We present new algorithms using structured matrix methods for manipulating polynomials expressed in generalized form, that is, relative to a polynomial basis $\{p_i(x)\}$ satisfying a three-term recurrence relation. This includes computing the Euclidean remainders as well as the greatest common divisor of two polynomials $u(x)$ and $v(x)$ of degrees less than $n$. The computational schemes involve solving linear systems with nested Hankel matrices represented in the given generalized basis and they reach the optimal sequential complexity of $O(n^2)$ arithmetical operations.

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