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

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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.