求解病态线性方程组的误差转移法和增广方程组法的机理及算法改进

The Mechanism and Improvement of Error Transferring Method and Augmented System Method for Solving Ill-conditioned Linear Equations

为获得病态线性方程组的高精度解,建立了一种优化模型,其最优解等价于早先提出的误差转移法和增广方程组法;指出后两者的本质机理是通过极小化解的模来近似极小化解的误差.为使算法适用于数据有污染的情况,进行了正则化改造.证明了新算法理论上与Tikhonov正则化等价.但当正则化参数趋于0时,目标函数的不同使得两者性能迥异,新算法可直接用于数据无污染的情况,而后者仍需选取合适的正则参数.数值算例验证了算法的有效性.

In order to obtain high-precision solution of ill-conditioned linear equations,an optimization model is established.Its optimal solution is equivalent to the results of error transferring method and augmented system method proposed earlier.This explains the essential mechanism of the latter two is to approximately minimize the error by minimizing the modulus of the solution.To be applied to the ill-conditioned problem when the initial data is polluted,the algorithm is improved by regularization.It is proved that the new algorithm is theoretically equivalent to Tikhonov regularization.However,when the regularization parameter tends to 0,the performance of the two methods is quite different due to the difference of their objective functions.The new algorithm can be directly used in the case of data pollutionfree,while the latter still needs to select an appropriate regularization parameter.Numerical examples demonstrate the effectiveness of the new algorithm.