Modern control theory is well known to strongly rely on numerical analysis, in particular for reducing matrices to simpler forms. It is less well known that, reciprocally, dynamical systems can play a significant role in the design and analysis of efficient algorithms for matrix computation, and for numerical analysis and optimization problems in general.
Interest in studying continuous-time flows related to computational problems started in the early eighties, when iterates of the unshifted QR algorithm (a celebrated method for eigenvalue computation) were shown to be the unit-time samples of the solution trajectory of a particular ODE (Lax-pair equation). This work sparked extensive research on using dynamical systems to solve linear algebraic problems. Notably, Brockett (1991) showed that the tasks of diagonalizing a matrix, linear programming, and sorting, could all be solved by continuous-time systems.
In this talk, several interesting features of continuous-time algorithms will be discussed, and analysis and design techniques will be illustrated on a few practical examples.
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