The nearest polynomial with a given zero, revisited

Abstract

In his 1999 SIGSAM BULLETIN paper [7], H. J. Stetter gave an explicit formula for finding the nearest polynomial with a given zero. This present paper revisits the issue, correcting a minor omission from Stetter's formula and explicitly extending the results to different polynomial bases.Experiments with our implementation demonstrate that the formula may not after all, fully solve the problem, and we discuss some outstanding issues: first, that the nearest polynomial with the given zero may be identically zero (which might be surprising), and, second, that the problem of finding the nearest polynomial of the same degree with a given zero may not, in fact, have a solution. A third variant of the problem, namely to find the nearest monic polynomial (given a monic polynomial initially) with a given zero, a problem that makes sense in some polynomial bases but not others, can also be solved with Stetter's formula, and this may be more satisfactory in some circumstances. This last can be generalized to the case where some coefficients are intrinsic and not to be changed, whereas others are empiric and may safely be changed. Of course, this minor generalization is implicit in [7]; This paper simply makes it explicit.

The nearest polynomial with a zero in a given domain

“…There are a lot of studies on the locations of the zeros of a polynomial in research on symbolic-numeric algorithms, for example, [6][7][8][9][10]12,15,17,[19][20][21]. In this paper, the l p -norm of a polynomial is the p-norm of the vector of coefficients of the polynomial with respect to a given basis.…”

Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

“…Zhi, Wu, Noda, Kai, Rezvan and Corless [31,30,23] generalize the formula to roots with given multiplicities. In [8] In [24,21] Stetter's multivariate (complex) formula is applied to the important problem of computing the nearest consistent polynomial system, with zeros of a minimum given multiplicity, and a different proof via generalized Lagrangian interpolation is given.…”

Rounding coefficients and artificially underflowing terms in non-numeric expressions

“…In [33] and in [27] we find polynomials being adjusted so as to be zero at given places (an extension in [26] to weighted norms is also of interest). If the measure of 'beauty' is how well the polynomials fit the given zero, then again the approach of that construction is similar to the approach of this present paper.…”