Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schoenhage in 1982. The main drawbacks of Pan's method are that it is quite involved and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANEWDSC can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.
The standard Descartes method has a complexity of @math for isolating the real roots of an integer polynomial of degree @math with coefficients bounded by @math in absolute value, see @cite_17 . The size of the recursion tree is @math , and @math arithmetic operations on numbers of bitsize @math need to be performed at each node. For @math , these bounds are tight, that is, there are examples where the recursion tree has size @math and the numbers to be handled grow to integers of length @math bits.
@cite_17
2158622794
We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = Eni=0 aiXi with integer coefficients |ai| < 2L, this yields a bound of O(n(L + logn)) on the size of recursion trees. We show that this bound is tight for L = Ω(logn), and we use it to derive the best known bit complexity bound for the integer case.