A recursive algorithm for the incomplete partial fraction decomposition (Q1093310)

Given polynomials \(P_{m+n-1}\), \(D_ m\), \(E_ n\) (where the subscripts denote the degree), the paper is concerned with improving the algorithms of constructing polynomials \(Q_{n-1},R_{m-1}\) in \(P_{m+n-1}=Q_{n- 1}D_ m+E_ nR_{m-1},\) originally described by \textit{P.Henrici} [Z. Angew. Math. Phys. 22, 751-755 (1971; Zbl 0247.65035)]. This algorithm may suffer from numerical instability, since it encompasses the solution of an ill-conditioned system of algebraic equations. The algorithm described here is recursive; theoretical arguments and comparative examples show that the new algorithm is faster with only \(3mn+O(m^ 3)\) multiplications for \(m\ll n\) (compared with \((m+n)^ 3/3\) in the Henrici's one) and less inclined to numerical instability.