Let
M(
y) be a matrix whose entries are polynomial in
y, λ(
y) and υ(
y) be a set of eigenvalue and eigenvector of
M(
y). Then, λ(
y) and υ(
y) are algebraic functions of
y, and λ(
y) and υ(
y) have their power, series expansions
λ(
y)=β
0+β
1y+…+β
kyk+…(β
j∈
C), (1)
υ(
y)=γ
0+γ
1y+…+γ
kyk+…(γ
j∈
Cn), (2)
provided that
y=0 is not a singular point of λ(
y) or υ(
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, β
k and γ
k, using lower degree coefficien β
j and γ
j(
j=0,1,…,
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 β
k and γ
k, 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 γ(
y) or υ(
y)), and presents an algorithm to estimate the accuracy of the computed power series β
i, γ
j 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.
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