A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

Starting from a quadratic invariant manifold in terms of the residual vector {\textbf r}={\textbf B}{\textbf x}-{\textbf b} for an n -dimensional ill-posed linear algebraic equations system {\textbf B}{\textbf x}={\textbf b} , we derive an ODEs system for {\textbf x} which is equipped with a state feedback controller to enforce the orbit of the state vector {\textbf x} on a specified manifold, whose residual-norm is exponentially decayed. To realize the above idea we develop a very powerful implicit scheme based on the novel Lie-group method to integrate the resultant differential algebraic equation (DAE). Through numerical tests of inverse problems we find that the present Lie-group DAE algorithm can significantly accelerate the convergence speed, and is robust enough against the random noise.