The Jordan Structure of Residual Dynamics Used to Solve Linear Inverse Problems

Abstract

With a detailed investigation of n linear algebraic equations Bx = b, we find that the scaled residual dynamics for y2 S n 1 is equipped with four struc- tures: the Jordan dynamics, the rotation group SO(n), a generalized Hamiltonian formulation, as well as a metric bracket system. Therefore, it is the first time that we can compute the steplength used in the iterative method by a novel algorithm based on the Jordan structure. The algorithms preserving the length of y are developed as the structure preserving algorithms (SPAs), which can significantly accelerate the convergence speed and are robust enough against the noise in the numerical solutions of ill-posed linear inverse problems.