A pre-processing hybrid algorithm for solving ill-conditioned linear equations

When direct solution and iterative method are devoted to solving for the linear equations, it may often lead to severe distortion of numerical solution because of the large condition number of ill-conditioned linear equations. Aiming at this problem, a pre-processing hybrid algorithm is proposed in this paper. A pre-processing method based on pivot element weighting is used for reducing condition number of coefficient matrix and the proof for reduction of condition number is shown. In a hybrid algorithm based on pivot weight, numerical solution of iterative algorithm is served as initial value. Then the initial value is optimized further by a least squares method. Finally, the algorithm is tested on large-scale over-determined linear equations which are linearized from the model of combinable magnetic resonance log. Contrast experimental results show that the algorithm proposed in this paper is able to reduce the error of the numerical solution and speed up convergence.