On Homogeneous Linear Recurrence Relations and Approximation of Zeros of Complex Polynomials

Abstract. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a − p(a)Dm−2(a)/Dm−1(a)}, where Dm(a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm(a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m ≥ 2 and an appropriate input a0, the fixed-point iterates ak+1 = Bm(ak) converge to a simple root having order m. The Basic Sequence corresponds to the pointwise evaluation of the Basic Family. We present several fractal images that confirm the theoretical convergence results: as m increases, the basins of attractions to the roots, as computed with respect to the iteration function Bm(z), rapidly converge to the Voronoi regions. Thus the regions with chaotic behavior rapidly shrink to the boundaries of the Voronoi regions. The interplay between the Basic Sequence and the Basic Family result in new algorithms for approximation of polynomial roots. Also as a byproduct of the analysis we give determinantal representation of solution of homogeneous linear recurrence relations, easily computable bounds on the modulus of terms of such recurrence relations, and fast computation of determinant of certain Toeplitz matrices. Finally, we prove a relationship between the Basic Sequence and the iterates of Bernoulli Method for approximation of extreme roots.Technical report DCS-TR-41