方程奇异值分解法在秩亏自由网平差中的应用

第45卷第13期 人 民 长 江 Vol〃45 No〃 13

2 0 1 4年7月 Yangtze iver July  2014

文章编号: 10014179(2014) 13003703

奇异值分解法在秩亏自由网平差中的应用

李行洋

(湖北水利水电职业技术学院湖北武汉430070)

摘要:在经典测量平差中按间接平差法列出的误差方程系数阵为列满秩所得到的法方程系数阵为一个对称的

满秩矩阵法方程组具有唯一解。而当控制网中没有必要的起算数据时则按间接平差法所列出的误差方程系数

阵是非列满秩的其对应的法方程系数阵也是一个秩亏的奇异阵。探讨了秩亏自由网的法方程系数阵的一种解

算方法。

关 键 词:秩亏自由网;极小范数解;特征值;正交向量

中图法分类号: TV131 文献标志码: A

秩亏自由网的特点是法方程的系数阵非满秩。目  Nm =N(NN)  ( 5)前解算秩亏自由网的方法很多1  2 但大多数方法的 Nm满足

主要思路是将秩亏矩阵转化为满秩矩阵求逆计算工 NNmN=N (6)作十分困难。本文采用奇异值分解法对秩亏矩阵求 (NmN) T =NmN (7))式1 方法与公式 NN=UUT (8)t×t t×t t×t t×tt×t

设观测误差方程 其中为对角矩阵且它的3主4对角元素(奇异值)恰

V=BδX^+l 权为 ( 1) 有t0个非零正值及d个零值 。这t个元素就是对称

Pn×1 n×t t×1 n×01 n×n 矩阵NN的特征值按递减次序排列。U的列向量即为

式中N=BT PB为实对称阵W=BT Pl 。0 可以证明NN的特征值即为N的特征值的平方。

δX^TδX^=min (3) 2 精度评定

极小范数解不妨取 δX^(或X^)的协因数阵QX^X^为

收稿日期:20140420

作者简介:李行洋男讲师主要从事工程测量的教学及科研工作。Email: xiao8243@163〃 com

38 人 民 长 江 2014年



N(Q。

) 

) 。

(NQ

16

第13期 李行洋:奇异值分解法在秩亏自由网平差中的应用 39

结语

) 在秩亏自由网中由于法方程系数阵为实对

 

16 N1是唯一的。

(编辑:赵凤超)

Application of singular value decomposition in rank defect free network adjustment

LI Xingyang

(Hubei Wateresources Technical College Wuhan430072 China)

Abstract: In classical adjustment  the normal equations have a unique solution if the error equation coefficient matrix achievedby indirect adjustment is a full column rank  and the normal equation coefficient matrix achieved by the same method is a sym-metrical full rank matrix〃When there is no necessary initial data in control network  the error equation coefficient matrix by indi-rect adjustment is a non  full column rank and the corresponding normal equation is also a rank deficient singular matrix〃 Thepaper discusses the solution for normal equation coefficient matrix in the rank defect free network〃

Key words: rank defect free network; small norm solution; eigenvalue; orthogonal vector

櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅櫅

(上接第8页)

Preliminary analysis on benefit and risk of overimpoundment operation of reservoirs

CHEN KebingHUANG Taoxing

(College of WaterConservancy and HydropowerEngineeringHohai UniversityNanjing 210098 China)

Abstract: On the premise of safeguarding reservoir safety properly coordinating the contradiction between reservoir flood con-trol and comprehensive utilization of water resources to fully utilize flood resources can realize the maximum benefits ofreservoirs〃Taking a reservoir as case  this paper discusses the issues of improving the capacities of flood control and irrigation byflood regu- lation through engineering and management measures on the premise of safeguarding their safety puts forwardcorrelative calcula- tion method about economic benefit and analyzes the risk control of reservoir over  impoundmentoperation〃 The analysis and calculation results show that proper over  impoundment operation of reservoirs can achieve significantbenefits of flood control and power generation with potential risk; over  impoundment operation of reservoirs should be carried outunder the condition of fully understanding the water regime〃

Key words: flood; flood resources utilization; flood controlbenefit; irrigation benefit; risk control