Continued fraction expansions in terms of discrete integrable systems and their applications to systems identifications and the BCH-Goppa decoding

离散可积系统及其在系统识别和 BCH-Goppa 解码中的应用方面的持续分数展开

• 批准号: 12554004
• 负责人: NAKAMURA Yoshimasa
• 金额: $ 4.67万
• 依托单位: KYOTO UNIVERSITY (2001-2003)Osaka University (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12554004/
• 关键词: discrete integrable systems; continued fractions; Pade approximation; Painleve equations; Laplace transform; applied integrable systems; 離散可積分系; Laplace変数; 組合せ論的数; Laplace変換; 直交多項式; Painleve方程式; 相対論戸田方程式; 可積分系; アルゴリズム; BCH-Goppa復号法; 直交多頂式

There has not been known a continued fraction expansion of order O(N^2) for the Perron continued fraction, which emerges in the Carathe\'odory interpolation problem, such as the qd algorithm for the Chebyshev continued fraction. First Nakamura and coworkers, being based on the orthogonal polynomials on the unit circle, derived a new integrable system named the Schur flow which has a Lax representation given by the three terms recurrence relation. Secondly in terms of the discrete Schur flow they designed a new continued fraction expansion algorithm of order O(N^2) for the Perron continued fraction and its application to algorithm for computing zeros of certain algebraic equations. Consequently, the new correspondence1)classical orthogonal polynomials -Chebyshev continued fraction -Toda equation2)orthogonal polynomials on the unit circle -Perron continued fraction -Schur flowis revealed.They also considered the Thron continued fraction through the relativistic Toda equation having a Lax representation given by the three terms recurrence relation for the bi-orthogonal polynomials. An integrable discretization of the equation enable them to design a new continued fraction algorithm of order O(N^3) for the Thron fraction. This algorithm has an advantage that it computes the continued fraction for the case where the FG algorithm does not work.Nakamura showed that a Pad\'e approximation, namely, a continued fraction expansion of the Laplace transform of the Airy function can be computed in a pure algebraic manner.Each coefficients of the continued fraction is connected by the By\"acklund transformation of the second Painlev\'e equation PII, where one of the Lax pair is just the recurrence relation of orthogonal polynomials.

对于Perron连分式，还没有已知的O（N^2）阶的连分式展开式，其出现在Caribe 'odory插值问题中，例如Chebyshev连分式的qd算法。首先，中村等人基于单位圆上的正交多项式，导出了一个新的可积系统Schur流，它具有由三项递推关系给出的Lax表示。其次，利用离散Schur流，设计了Perron连分式的O（N^2）阶连分式展开算法，并将其应用于求解某些代数方程组的零点。从而揭示了1）经典正交多项式-Chebyshev连分式-户田方程2）单位圆上的正交多项式-Perron连分式-Schur流的新对应关系，并通过相对论性户田方程考虑了Thron连分式，该方程具有由双正交多项式的三项递推关系给出的Lax表示.一个可积的离散方程使他们能够设计一个新的连续分数算法的顺序为O（N^3）的Thron分数。该算法的优点在于，它计算FG算法不工作的情况下的连分数。中村表明，可以以纯代数的方式计算Pad\“e近似，即Airy函数的拉普拉斯变换的连分数展开。连分数的每个系数由第二Painlev\”e方程PII的By\“acklund变换连接，其中Lax对之一就是正交多项式的递推关系。

项目成果

中山功(I.Nakayama): "誤差関数の評価不等式(Estimating Inequalities for the Error Function)"NUCB Journal of Economics and Information Science(名古屋商科大学論集). Vol.48. 89-100 (2003)

I. Nakayama：“估计误差函数的不等式”NUCB 经济与信息科学杂志（名古屋商业大学）第 48 卷 89-100（2003 年）。

S.Louboutin, R.Okazaki: "Exponents of the ideal class groups of CM number fields"Math.Z.. Vol.243. 155-159 (2003)

S.Louboutin、R.Okazaki：“CM 数域的理想类群的指数”Math.Z.. Vol.243。

R.Okazaki: "Geometry of a cubic Thue equation"Publ. Math. Debrecen. Vol.61. 267-314 (2002)

R.Okazaki：“三次Thue方程的几何”Publ。

A.Mukaihira, Y.Nakamura: "Schur flow for orthogonal polynomials on the unit circle and its integrable discretization"Journal of Computational and Applied Mathematics. Vol.139. 75-94 (2002)

A.Mukaihira、Y.Nakamura：“单位圆上正交多项式的 Schur 流及其可积离散化”计算与应用数学杂志。

Y.Minesaki, Y.Nakamura: "A conservative numerical integration algorithm for the integrable Henon-Heiles system"Proceedings of Institute of Mathematics of NAS of Ukraine, Institute of Mathematics, Kyiv. Vol.I. 444-449 (2004)

Y.Minesaki，Y.Nakamura：“可积 Henon-Heiles 系统的保守数值积分算法”乌克兰国家科学院数学研究所论文集，基辅数学研究所。