The Jordan structure of residual dynamics used to solve linear inverse problems

With a detailed investigation of n linear algebraic equations Bx = b, we find that the scaled residual dynamics for y ∈ S^n-1 is equipped with four structures: 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.