An Extension Study On Ill-conditioning Of Surveying Data Processing

Generally,the ill-conditioning problem not only exists in Geodesy,such as global positioning system(GPS) data processing, geophysical inversion, observation deformation analysis,etc,but also its effects are very serious.When ill-conditioning problems are encountered,it is of great significance to the decline of ill-conditioning due to analysis from the root.The dissertation begins with the review of the main history of ill-conditioning research.And then it emphatically discusses and investigates the expansion of the traditional ill-conditioning study,concerning the spatial geometry analysis of the ill-conditioning problems and nonlinear ill-conditioning problems.This paper explores the ill-conditioning from spatial geometry, which overcomes the abstract and incomprehensible problem of numerical analysis.First,the ill-conditioning problem is discussed from the perspective of the variation in the volume and shape,including the constitution of the spatial geometric figure of corresponding matrices,the norm expression of the volume of multidimensional parallel polyhedron.Moreover,the relations have been proven between the volume size of the multidimensional parallel polyhedron and the collinearity.Second,the volume computation is introduced based on Singular Value Decomposition(SVD).The multidimensional rectangular polyhedron,generalizing the relations between it and the condition number,is rebuilt with SVD,which deepen and underpin the understanding of the ill-conditioning problem.Finally, the variation of multidimensional ellipse is used to determinate the number and location of multicollinearity.Therefore, this paper succeeds in endowing the determinant analysis, feature analysis and condition number with geometric meanings in the diagnostics and measurement of the multicollinearity .The ill-conditioning study is developed from the linear model to nonlinear model,which is a meaningful combination between ill-conditioning study and nonlinear theory.First,the four kinds of combination are introduced,which all are named nonlinear ill-conditioning problems.Second,the impact of ill-conditioning research,from two aspects of nonlinear algorithm ,is analysed.One is judgement of multicollinearity from nonlinear model approximate solution and another is nonlinear iterative algorithm owing to ill-conditionilng. Finally, stability is introduced due to the selection of a nonlinear model.