An E cient Algorithm for the Real Root and Symmetric Tridiagonal Eigenvalue Problems

Given a univariate complex polynomial f(x) of degree n with rational coe cients expressed as a ratio of two integers < 2, the root problem is to nd all the roots of f(x) up to speci ed precision 2 . In this paper we assume the arithmetic model for computation. We give an improved algorithm for nding a well-isolated splitting interval and for fast root proximity veri cation. Using these results, we give an algorithm for the real root problem: where all the roots of the polynomial are real. Our real root algorithm has time cost of O(n log n(logn+ log b)); where b = m + . Our arithmetic time cost is thus O(n log n) even in the case of high precision b n. This is within a small polylog factor of optimality, thus (perhaps surprisingly) upper bounding the arithmetic complexity of the real root problem to nearly the same as basic arithmetic operations on polynomials. The symmetric tridiagonal problem is: given an n n symmetric tridiagonal matrix, with 3n nonzero rational entries each expressed as a ratio of two integers < 2, to nd all the eigenvalues up to speci ed precision 2 . Using known e cient reductions from the symmetric tridiagonal eigenvalue problem to the real root problem, we also get an O(n log n(logn+log b)) arithmetic time bound for the symmetric tridiagonal eigenvalue problem.