A fraction-free unit-circle zero location test for a polynomial with any singularity profile

Alexander Lifshitz, Yuval Bistritz

Research output: Contribution to journalArticlepeer-review

Abstract

This paper presents a fraction-free (FF) version of the Bistritz test to determine the zero location (ZL) of a polynomial with respect to the unit circle. The test has the property that when it is invoked on a polynomial with Gaussian or real integer coefficients, it is an efficient integer algorithm completed without fractions over the respective integral domain. The test is not restricted to integers but remains integer preserving (IP) in all possible encounters of abnormalities and singularities. We define a symmetric subresultant polynomial sequence (SSPS) for the Sylvester matrix of two symmetric polynomials. We then show that the sequence of polynomials produced by the FF test coincides with the SSPS of its first two polynomials when the test is normal and the SSPS is strongly nonsingular, or else its polynomials match the non-singular subresultant polynomial and pass over intermediate gaps of singular subresultants in an IP and efficient manner. This relationship (interesting in its own right) is used to show that the test is IP and normally attains integers of minimal size.

Original languageEnglish
Pages (from-to)1420-1459
Number of pages40
JournalLinear and Multilinear Algebra
Volume67
Issue number7
DOIs
StatePublished - 3 Jul 2019

Keywords

  • 11C08
  • 11C20
  • 15B36
  • 93D99
  • Schur–Cohn problem
  • Sylvester resultant
  • discrete-time linear systems stability
  • immittance algorithms
  • integer algorithm
  • subresultants
  • symmetric polynomials

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