Enclosing clusters of zeros of polynomials

摘要

Lagrange interpolation and partial fraction expansion can be used to derive a Gerschgorin-type theorem that gives simple and powerful a posteriori error bounds for the zeros of a polynomial if approximations to all zeros are available. Compared to bounds from a corresponding eigenvalue problem, a factor of at least two is gained.The accuracy of the bounds is analyzed, and special attention is given to ensure that the bounds work well not only for single zeros but also for multiple zeros and clusters of close zeros.A Rouché-type theorem is also given, that in many cases reduces the bound even further.