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

摘要

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, f ∈ R[x] such that f has a multiple zero in I and ‖f - f‖∞, the infinity norm of the vector of coefficients of f - f, is minimal. First, we prove that if a nearest polynomial f exists, there is a nearest polynomial g ∈ R[x] such that the absolute value of every coefficient of f - g is ‖f - f‖∞ 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 g. Furthermore, we give a rational function whose value at α is ‖f - f‖∞.