Discrete hungry integrable systems — 40 years from the Physica D paper by W.W. Symes

The Toda equation is a famous integrable system studied in multiple fields, including mathematical physics and numerical computing. Forty years ago, Symes showed that the time-1 evolution in the Toda equation corresponds to the 1-step of the well-known QR algorithm whose target matrices are tridiagonal exponentials. The discrete Toda (dToda) equation proposed by Hirota is, in fact, just the recursion formula of the quotient-difference (qd) algorithm for computing tridiagonal eigenvalues. The discrete Lotka–Volterra (dLV) system describing predator–prey interactions is closely related to the dToda equation and can be used to compute tridiagonal matrices. In this paper, by focusing on relationships to Hessenberg eigenvalues, we summarize results for discrete hungry integrable systems that are extensions of the dToda equation and the dLV system. Our main approach is to utilize polynomial sequences with two types of discrete time. We simultaneously clarify the solutions to the discrete hungry integrable systems and their Bäcklund transformations. We then describe continuous analogues of the discrete hungry integrable systems. Moreover, we present desirable properties of practical algorithms based on the discrete hungry integrable systems.