Studying Diagnostics And Processing For Ill-conditioning And Algorithm For Solving Ill-conditioned Complex Linear Equations

The solves of ill-conditioned complex linear equations are the important issue in numerical calculation.We discuss the diagnosis and measure,preconditioning and the solving of ill-condition and than propose some processing methods and improvement algorithm on the basis of research history and current status of ill-condition problems.The ill-condition is defined and described qualitatively and the relationship of ill-condition with multicollinearity is discussed.In the methods of diagnosis and measure we also introduce feature analysis,condition number and condition index & variance decomposition which with perfect theory and point out their merit and demerit explicitly.Finally a new method which is used to eliminate the extra parameters of multicollinearity by combining condition index & variance decomposition with AdaptiveLasso is proposed.With it people can avoid the blindness and subjectivity when eliminating the extra parameters of multicollinearityThe basic idea of preconditioning is introduced and according to the trait of ill-conditioned complex linear equations the SSOR preconditioning matrix and two of its improvements are recommended.And than take symmetric and positive complex linear equations as an example,a new preconditioned complex-conjugate gradient which is suitable to solve the complex linear equations is put forward.Finally we also put forward a new preconditioned complex-biconjugate gradient by combining the improvement of SSOR preconditioning matrix with bi-conjugate gradient method,and it is proved higher precision and stability on the data experiment.The current main solutions of ill-condition system are classified into statistical estimation algorithm and swarm intelligence algorithm and than the usage and principle of each of them are also be analyzed.Finally we put forward a new hybrid algorithm by combining the complex-biconjugate gradient with simulated annealing algorithm which have complementary traits,and it is proved higher precision and practicability on the data experiment.