Partial Fraction Decomposition in C(z) and Simultaneous Newton Iteration for Factorization in C[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. Scho nhage (``The Fundamental Theorem of Algebra in Terms of Computational Complexity,'' Technical Report, Univ. Tu bingen, 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.

Root refinement for real polynomials using quadratic interval refinement

“…We can further reduce the complexity of our method using the technique of fast approximate multipoint evaluation [37,22,24]. With that approach, performing one refinement step on all isolating intervals simultaneously has the same cost (up to logarithmic factors) as a single refinement step with classical evaluation.…”

Nearly optimal computations with structured matrices

“…Hankel matrices H = (hi+j ) Kirrinnis' results to the solution of a Cauchy linear system of equations (which unlike [11] covers rational interpolation) and in Section 7.2 to the solution of Trummer's celebrated problem [9], [10], [6], having important applications to mechanics (e.g., to particle simulation) and representing the secular equation, which is the basis for the MPSolve, the most efficient package of subroutines for polynomial rootfinding [3].…”

“…Now we can use [11,Theorem 3.9] choosing λ = ℓ + τ1 + 2(n+m)ρ to guarantee f mod pj − fj 1 = fj − fj 1 ≤ 2 −ℓ . The complexity of the procedure is OB(µ((m + n lg n)(ℓ + τ1 + (m + n)ρ))).…”

“…We also wish to cite the papers [25] and [13], although their results have been superceded in the advanced work of 1998 by Kirrinnis, [11], apparently still not sufficiently well known. Namely in the process of studying approximate partial fraction decomposition he has estimated the Boolean complexity of the multipoint evaluation, interpolation, and the summation of rational functions.…”

“…In both representations the computational tasks and the solution algorithms are equivalent, and so the results of [11] for partial fraction decomposition can be extended to most although not all of these tasks. By using both representations, however, we make our analysis more transparent.…”