ASCM 2007: Computer Mathematics pp 32-41

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

• Hiroshi Sekigawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)

## 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 $$\|f - \tilde{f}\|_\infty$$.

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