The Nearest Real Polynomial with a Real Multiple Zero in a Given Real Interval

Abstract

Given f ∈ ℝ[x] and a closed real interval I, we provide a rigorous method for finding a nearest polynomial with a real multiple zero in I, that is, $\tilde{f}\in\mathbb{R}[x]$ such that $\tilde{f}$ has a multiple zero in I and $\|f - \tilde{f}\|_\infty$, the infinity norm of the vector of coefficients of , is minimal. First, we prove that if a nearest polynomial exists, there is a nearest polynomial $\tilde{g}\in\mathbb{R}[x]$ such that the absolute value of every coefficient of $f-\tilde{g}$ is $\|f - \tilde{f}\|_\infty$ with at most one exceptional coefficient. Using this property, we construct h ∈ ℝ[x] such that a zero of h is a real multiple zero α ∈ I of $\tilde{g}$. Furthermore, we give a rational function whose value at α is $\|f - \tilde{f}\|_\infty$.