A new algorithm for solving ill-conditioned linear systems

A new algorithm is presented for solving ill-conditioned linear equations. This method is radically different from conventional methods. We improve the condition of the linear equation first and then solve the well-conditioned one. The approximate solution is modified to get the exact solution. Numerical simulations show that the proposed method is effective enough to be applied to the practical finite element problems. Mathematical soundness of the algorithm is also shown. Several implementation techniques are discussed. It can be used not only for solving ill-conditioned equations but also for inverse problems and many other areas.