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

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概要

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• 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 expansionsλ(y)=β0+β1y+…+βkyk+…(βj∈C), (1)υ(y)=γ0+γ1y+…+γkyk+…(γj∈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 Newtons 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 coefficien βj and γj(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 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 γ(y) or υ(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 Cauchys integral formula and numerical integrations. We present an efficient implementation of the algorithm that utilizes Newtons method. We also present a modification of Newtons 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.

• （社）電子情報通信学会の論文
• 2008-08-01

著者

• Yamaguchi Tetsu

  Cybernet Systems Co. Ltd

• KITAMOTO Takuya

  Faculty of Education, Yamaguchi University

• Kitamoto Takuya

  Faculty Of Education Yamaguchi University

関連論文

• The Optimal H_∞ Norm of a Parametric System Achievable by an Output Feedback Controller
• On the Check of Accuracy of the Coefficients of Formal Power Series
• The Optimal H_∞ Norm of a Parametric System Achievable Using a Static Feedback Controller(Systems and Control)
• Extension of the Algorithm to Compute H_∞ Norm of a Parametric System
• Efficient Computation of the Characteristic Polynomial of a Polynomial Matrix
• On Efficient Computations of Approximate Roots
• Accurate Computation of a High Degree Coefficient of a Power Series Root(Algorithms and Data Structures)
• On Computation of Approximate Eigenvalues and Eigenvectors
• On Computation of a Power Series Root with Arbitrary Degree of Convergence
• Computation of the Peak of Time Response in the Form of Formal Power Series(Systems and Control)

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