A new approach to numerical algorithms in terms of integrable systems

Abstract

Almost four decades passed after the discovery of solitons and infinite dimensional integrable systems. The theory of integrable systems has had great impact to wide area in physics and mathematics. An approach to numerical algorithms in terms of integrable systems is surveyed. Some integrable systems of Lax form describe continuous flows of efficient numerical algorithms, for example, the QR algorithm and the Jacobi algorithm. Discretizations of integrable systems in tau-function level enable us to formulate algorithms for computing continued fractions such as the qd algorithm and the discrete Schur flow. A new singular value decomposition (I-SVD) algorithm is designed by using a discrete integrable system defined by the Christoffel-Darboux identity for orthogonal polynomials.