On the Check of Accuracy of the Coefficients of Formal Power Series

Let M(y) be a matrix whose entries are polynomial in y, λ(y) and v(y) be a set of eigenvalue and eigenvector of M(y). Then, λ(y) and v(y) are algebraic functions of y, and λ(y) and v(y) have their power series expansions λ(y) = β0 + β1y + ₀ + βkyk + ₀ (βj e C),            (1) v(y) = γ0 + γ1y + ₀ + γkyk + ₀ (γj e Cn),            (1) provided that y = 0 is not a singular point of λ(y) or v(y). Several algorithms are already proposed to compute the above power series expansions using Newton's method (the algorithm in [4]) or the Hensel construction (the algorithm in [5],[12]). The algorithms proposed so far compute high degree coefficients βk and γk, using lower degree coefficients βj and γj (j = 0,1,₀,k-1). Thus with floating point arithmetic, the numerical errors in the coefficients can accumulate as index k increases. This can cause serious deterioration of the numerical accuracy of high degree coefficients βk and γk, and we need to check the accuracy. In this paper, we assume that given matrix M(y) does not have multiple eigenvalues at y = 0 (this implies that y = 0 is not singular point of λ(y) or v(y)), and presents an algorithm to estimate the accuracy of the computed power series βi,γj in (1) and (2). The estimation process employs the idea in [9] which computes a coefficient of a power series with Cauchy's integral formula and numerical integrations. We present an efficient implementation of the algorithm that utilizes Newton's method. We also present a modification of Newton's method to speed up the procedure, introducing tuning parameter p. Numerical experiments of the paper indicates that we can enhance the performance of the algorithm by 12 ~ 16%, choosing the optimal tuning parameter p.