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

对于Perron连分式，还没有已知的O（N^2）阶的连分式展开式，这种展开式出现在Carathe 'odory插值问题中，如Chebyshev连分式的qd算法。首先，Nakamura等人基于单位圆上的正交多项式，导出了一种新的可积系统Schur流，该系统具有由三项递推关系给出的Lax表示。其次，针对离散舒尔流，设计了一种新的O（N^2）阶的Perron连分式展开算法，并将其应用于某些代数方程的零计算算法。由此，揭示了新的对应关系：1)经典正交多项式-Chebyshev连分数-Toda方程2)单位圆上正交多项式-Perron连分数-Schur流。他们还考虑了Thron连分式，通过相对论Toda方程，该方程具有由双正交多项式的三项递归关系给出的Lax表示。通过对方程的可积离散化，他们设计了一种新的O（N^3）阶的连分式算法。这个算法有一个优点，它可以在FG算法不起作用的情况下计算连分数。Nakamura证明了Pad\'e近似，即Airy函数的拉普拉斯变换的连分式展开式可以用纯代数方式计算。连分式的每个系数由第二个painleve方程PII的by acklund变换连接起来，其中Lax对中的一个就是正交多项式的递归关系。

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.