Numer. Math. 20, 139---148 (t972) 9 by Springer-Verlag 1972 A posteriori Error Bounds for the Zeros of a Polynomial Martin Gutknecht Seminar flit Angewandte Mathematik, Eidgen6ssische Technische Hochschule Ziirich Received 14 June, 197t Summary. An algorithm for the computation of error bounds for the zeros of a polynomial is described. This algorithm is derived by applying Rouch6's theorem to a Newton-like interpolation formula for the polynomial, and so it is suitable in the case where the approximations to the zeros of the polynomial are computed successively using deflation. Confluent and clustered approximations are handled easily. However bounds for the local rouding errors in deflation, e.g. in Homer's scheme, must be known. In practical application the method can, especially in some ill-conditioned cases, compete with other known estimates. 1. Introduction B6rsch-Supan [3] and Smith [t6] apply Rouch6's theorem to compute error bounds for approximate zeros of a polynomial. Their error bounds result as the zero of real auxiliary functions, which are derived from Lagrange's interpolation formula. Also Smith [16] indicates briefly how Rouch6's theorem can be applied to Newton's instead of Lagrange's interpolation formula. In this paper an algo- rithm is derived which utilizes his technique and which handles rounding errors as in Wilkinson [t9]. This algorithm is suitable in the case where approximations to the zeros are computed successively using deflation. The bounds for simple, confluent, and clustered approximations are all calculated from the same formulas, which facilitates programming. This and the lack of instability is an advantage in comparison with the algorithms of B6rsch-Supan [3 ~ and Smith [161, and similar other methods [2, 4, t3, t4, t 51, which usually give slightly better results at the price of greater computational expenses. On the other hand our estimate is much better than those of Nickel [t0, ti], who uses classical formulas of Fekete and Ballieu. I am indebted to Dr. Brian T. Smith for an indication of this method and for many hints for its implementation. 2. -Theoretical Background Suppose that PN is a given complex monic polynomial of degree N of the variable z and that z N is an approximation to one of its zeros. If we divide PN by z- zN there results a polynomial P N-1 of degree N- 1 and the fraction P N (zN)/ (z--zN), which we delete. By iteration of this process called deflation we get N monic polynomials p~ (k=N, N--! ..... 1) with respective degree k. More exactly, taking round-off into account, we denote by Pk the actually computed