基于共轭梯度搜索的病态问题处理方法

在最小二乘平差准则基础上，把病态平差问题转化为无约束的二次规划问题，并利用优化理论分析病态对平差解的影响。通过共轭梯度搜索算法在可行域中寻找最优步长因子，自动寻找最速下降方向，并给出迭代初值的设置方法。分析近似计算中病态问题与局部最优解的关系，讨论局部最优解的快速迭代方法，并通过实例验证算法的有效性，计算迭代的速度。由于整个过程没有对法方程系数矩阵进行求逆计算，该算法可用于处理大规模系数矩阵高病态的平差问题。

A Processing Method of Ill-Posed Problems Based on Conjugate Gradient Search

In this paper,based on the least square adjustment criterion, the ill-posed adjustment problem is transformed into an unconstrained quadratic programming problem. At the same time, the influence of ill-posed problems on the solution of the adjustment is analyzed by using optimization theory. The conjugate gradient search algorithm is used to find the optimal step size factor in the feasible region and automatically ascertain the steepest descent direction. The setting method of iterative initial value is given, and we analyze the relationship between ill-posed problems and local optimal solutions in approximate calculation, which gives a fast-iterative method of local optimal solution. We also analyze the validity of the algorithm and the speed of iteration through two examples. The algorithm can be used to deal with the highly ill-posed adjustment problem of large-scale coefficient matrices becausethe inverse matrix about the coefficient matrix of normal equation is not calculated in the whole calculation process.