求解大规模最小二乘问题的两种斜方向的Gauss-Seidel方法

TWO GAUSS-SEIDEL METHODS WITH OBLIQUE DIRECTION FOR SOLVING LARGE-SCALE LEAST SQUARES PROBLEMS

基于贪婪准则和最大距离准则选择系数矩阵工作列的策略,提出两种求解大规模超定不相容线性系统的斜方向的Gauss-Seidel方法,即斜方向的贪婪随机Gauss-Seidel(GRGSO)方法和斜方向的快速最大距离Gauss-Seidel(FMDGSO)方法.当系数矩阵是列满秩时,理论表明这些方法收敛到线性系统的唯一的最小二乘解。特别是当矩阵A的列接近线性相关时,数值结果表明这些方法在求解性能方面比传统的Gauss-Seidel型方法更具优势.

Two Gauss-Seidel type methods with oblique direction for solving large-scale overdetermined inconsistent linear systems,namely,the greedy random Gauss-Seidel method with oblique direction(GRGSO)and the fast max-distance Gauss-Seidel method with oblique direction(FMDGSO),are proposed with the use of the strategy of selecting the working sequence of the coefficient matrix based on the greedy criterion and the max-distance criterion respectively.When the coefficient matrix is of full column rank,the theoretical results show that the two methods converge to the unique least square solution of the linear system.Particularly,when the columns of matrix A are close to linearly correlated,the numerical results show that the two methods have more advantages than the existing Gauss-Seidel method in solving performance.