A statistical analysis of the numerical condition of multiple roots of polynomials

Componentwise and normwise condition numbers of an m-tuple root x0 of a polynomial p(x) that are appropriate for measurement and experimental inaccuracies are derived. These new condition numbers must be compared with the established condition numbers, which are appropriate for quantifying the effect of roundoff errors due to floating point arithmetic. It is shown that the condition numbers that are derived in this paper may be considered average case (as opposed to worst case) because extensive use is made of the expected values of random variables and functions of random variables. Specifically, it is assumed that each coefficient of p(x) is perturbed by an independent zero mean Gaussian random variable, and a measure of the condition of x0 is defined as the ratio of the expected value of its relative error to the expected value of the relative error in the coefficients of p(x), defined in both the componentwise and normwise forms. It is shown that this distinction between the componentwise and normwise condition estimates is important because they may differ by several orders of magnitude, depending on the coefficients of the polynomial. The cause of ill-conditioning of multiple roots is considered and it is shown that the situations m = 1 and m > 1 must be treated separately. Computational experiments that illustrate the theoretical results are presented.