| 修正的Gram-Schmidt正交化广义逆平差方法 |
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| 引用本文： | 罗三明,薄万举,黄曲红,王西宁.修正的Gram-Schmidt正交化广义逆平差方法[J].武汉大学学报(信息科学版),2012,37(2):174-177. |
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| 作者姓名： | 罗三明 薄万举 黄曲红 王西宁 |
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| 作者单位： | 1. 中国地震局第一监测中心,天津市耐火路7号,300180
2. 国家测绘局第一大地测量队,西安市测绘路4号,710054 |
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| 基金项目： | 地震行业科研专项重大资助项目，国家测绘局科技创新资助项目，中国地震局第一监测中心青年基金联合资助项目 |
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| 摘 要： | 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。
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| 关 键 词： | 修正的Gram-Schmidt正交化 线性方程组 秩亏系数阵 广义逆 最小二乘极小范数解 |
A Method of Generalized Inverse Adjustment Based on Improved Gram-Schmidt Orthogonalization |
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| LUO Sanming,BO Wanju,HUANG Quhong,WANG Xining.A Method of Generalized Inverse Adjustment Based on Improved Gram-Schmidt Orthogonalization[J].Geomatics and Information Science of Wuhan University,2012,37(2):174-177. |
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| Authors: | LUO Sanming BO Wanju HUANG Quhong WANG Xining |
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| Institution: | 1 First Crust Deformation Monitoring and Application Center,CEA,7 Naihuo Road,Tianjin 300180,China)(2 First Geodetic Surveying Brigade,SBSM,4 Cehui Road,Xi’an 710054,China) |
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| Abstract: | 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. |
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| Keywords: | improved Gram-Schmidt orthogonalization linear equation-group rank-defect coefficient matrix generalized inverse minimal norm least square solutions |
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