Partial Fraction Decomposition in [formula omitted](z) and Simultaneous Newton Iteration for Factorization in [formula omitted][z

Abstract

The subject of this paper is fast numerical algorithms for factoring univariate polynomials with complex coefficients and for computing partial fraction decompositions (PFDs) of rational functions in C(z). Numerically stable and computationally feasible versions of PFD are specified first for the special case of rational functions with all singularities in the unit disk (the “bounded case”) and then for rational functions with arbitrarily distributed singularities. Two major algorithms for computing PFDs are presented: The first one is an extension of the “splitting circle method” by A. Schönhage (“The Fundamental Theorem of Algebra in Terms of Computational Complexity,” Technical Report, Univ. Tübingen, 1982) for factoring polynomials in C[z] to an algorithm for PFD. The second algorithm is a Newton iteration for simultaneously improving the accuracy of all factors in an approximate factorization of a polynomial resp. all partial fractions of an approximate PFD of a rational function. Algorithmically useful starting value conditions for the Newton algorithm are provided. Three subalgorithms are of independent interest. They compute the product of a sequence of polynomials, the sum of a sequence of rational functions, and the modular representation of a polynomial. All algorithms are described in great detail, and numerous technical auxiliaries are provided which are also useful for the design and analysis of other algorithms in computational complex analysis. Combining the splitting circle method with simultaneous Newton iteration yields favourable time bounds (measured in bit operations) for PFD, polynomial factoring, and root calculation. In particular, the time bounds for computing high accuracy PFDs, high accuracy factorizations, and high accuracy approximations for zeros of squarefree polynomials are linear in the output size (and hence optimal) up to logarithmic factors.