修正的Gram-Schmidt正交化广义逆平差方法

A Method of Generalized Inverse Adjustment Based on Improved Gram-Schmidt Orthogonalization

直接从条件方程或误差方程系数阵入手,利用修正的Gram-Schmidt正交化过程对系数阵进行三角分解,实现最小二乘求解,导出了基于修正的Gram-Schmidt正交化过程求解系数阵广义逆的数学公式和计算步骤,给出了通过广义逆表示的未知数解向量及其协因数阵的数学表达式。计算过程不仅避免了对矩阵的求逆,并从理论上解决了Gram-Schmidt正交化方法由于舍入误差的影响表现出的数值不稳定性问题,从而很好地解决了具有秩亏系数阵方程组解的不唯一性。算例结果表明,基于修正的Gram-Schmidt正交化方法可以处理包括秩亏阵在内的任意矩阵;在处理不设起算数据的变形监测网观测数据时,能够方便地获得其经典解、伪逆解或拟稳解,而不需要重复计算。

Starting directly with coefficient matrix of condition equation or error equation,the least square solution by triangulation decomposition on coefficient matrix is carried on with improved Gram-Schmidt orthogonalization procedure.Then,the math formula and the calculation steps of solving generalized inverse matrix on improved Gram-Schmidt algorithm are deduced.The unknown solution vectors and the mathematical expression of the variance-covariance matrix are given through the generalized inverse expression.Two examples are used to verify its effect,and the results show that the modified Gram-Schmidt orthogonal method can process any matrix including rank defect array.