Approximate Greatest Common Divisors and Polynomials Roots

This lecture will show by example some of the problems that occur when the roots of a polynomial are computed using a standard polynomial root solver. In particular, polynomials of high degree with a large number of multiple roots will be considered, and it will be shown that even roundoff error due to floating point arithmetic, in the absence of data errors, is sufficient to cause totally incorrect results. Since data errors are usually larger than roundoff errors (and fundamentally different in character), the errors encountered with real world data are significant and emphasise the need for a computationally robust polynomial root solver. These computational results will be quantified by considering the componentwise and normwise condition numbers, and the componentwise and normwise backward errors, of a root of a polynomial, and it will be shown that the forward error is not a practical error measure. The equation that links the forward error, backward error and condition number of a root of a polynomial will be considered.