The nearest polynomial with a zero in a given domain

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For a real univariate polynomial f and a closed domain D ⊂ C whose boundary C is represented by a piecewise rational function, we provide a rigorous method for finding a real univariate polynomial over(f, ̃) such that over(f, ̃) has a zero in D and {norm of matrix} f - over(f, ̃) {norm of matrix} is minimal. First, we prove that if a nearest polynomial exists, there is a nearest polynomial over(f, ̃) such that the absolute value of every coefficient of f - over(f, ̃) is {norm of matrix} f - over(f, ̃) {norm of matrix} with at most one exception. Using this property and the representation of C, we reduce the problem to solving systems of algebraic equations, each of which consists of two equations with two variables.

Original languageEnglish
Pages (from-to)282-291
Number of pages10
JournalTheoretical Computer Science
Issue number2
Publication statusPublished - 17 Dec 2008



  • Perturbation
  • Polynomial
  • Zero
  • l-norm

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