用拟准检定法和岭估计解算病态污染方程

RESOLUTION OF THE ILL-CONDITIONED AND POLLUTED EQUATION WITH QUASI-ACCURATE DETECTION AND RIDGE ESTIMATE

针对法方程呈现病态且观测值受到污染而不严格服从正态分布的情况 ,探讨运用拟准检定法抗御粗差 ,用岭估计改善法方程的病态性。拟准检定法是根据真误差与观测值的确定关系式 RΔ =- RL求解真误差估值Δ。在 A或 ATA病态的情况下 ,直接求 R是不稳定的。对 A作奇异值分解计算 R,可避免对病态矩阵直接求逆。由于 R是秩亏的 ,通过附加“拟准观测真误差估值范数极小”的条件来求解真误差估值。拟准检定法与岭估计两者的结合 ,一般情况下有效地剔除了粗差 ,而且改善了方程的病态性 ,计算简单、快捷 ,结果比较理想。

This paper discusses the resistance of gross errors with quasi accurate detection and the improvement of ill conditioned equation with ridge estimate when the equation is ill conditioned and polluted.The quasi accurate detection is the method that can resolve the real error estimators according to the following relation between real errors and the observations:RΔ=-RL.However,the direct calculation of R is not steady due to the ill condition of matrix A or A T A.In order to calculate the matrix R,matrix A is processed with SVD method in advance.In addition,because of the rank deficiency of matrix R,this paper introduces the idea that the real error estimators are resolved by adding the restrained condition that the norm of the real errors of the quasi accurate observations (QAO) is minimum.By combining quasi accurate detection with ridge estimate,the gross errors can be eliminated effectively and ill condition of equation can be improved.The resolution is simple and the result is perfect comparatively.