A New Method Of Surveying Adjustment By Using Generalized Inverses Based On {1}-Inverse

Surveying adjustment based on generalized inverse matrix began in 1971,when Mittermayer proposed the pseudo inverse solution of rank deficient free network adjustment.After half a century of development,the application of generalized inverse in adjustment field has been further extended to using generalized inverse as a tool to derive adjustment formulas of elements,conditions,elements with condition,conditions with parameter,quasi-stable adjustment,least square filtering and collocation,and also to some new adjustment theories and methods.The surveying adjustment based on generalized inverse can not only avoid using normal equations and simplify the derivation process of the adjustment formula,but also provide a simple solution formula that is convenient for calculation.Therefore,it is very convenient,appropriate and effective to use generalized inverse matrix to resolve surveying adjustment.Surveying adjustment based on generalized inverse matrix inevitably involves the calculation of generalized inverse.However,the existent algorithm of generalized inverse used in surveying adjustment is relatively single and the computation load is also big,with the calculation complicated,which means that computer programming is also complicated.The difficulty of calculation limits the further application and development of generalized inverse in the surveying adjustment.This thesis presents a new algorithm for generalized inverses based on{1}-Inverse and applies it to the solution of surveying adjustment.In addition,this thesis presents new generalized inverse solutions for the five adjustment models,including elements,conditions,elements with condition,conditions with parameter and free networks and compares the efficiency and accuracy of the new generalized inverse solutions with those of the existing ones by analysis of computational complexity and simulation of numerical experiment.The main research achievements of this thesis are as follows:（1）In addition to weighted least squares inverseA_l（P）^-,weighted minimum norm inverseA_m（PX）^-and the weighed pseudo-inverseA_PPX⁺,the three most commonly used generalized inverses in current surveying adjustment,more types of generalized inverses,namely generalized inverses based on{1}-Inverse are introduced into the surveying adjustment as a tool for derivation,calculation and expression of adjustment formulas and accuracy estimation to combine generalized inverses as a mathematic tool more deeply with surveying adjustment.（2）In this thesis,a new algorithm is raised for generalized inverses based on{1}-Inverse.Compared with the existing algorithms for generalized inverses in mathematics and surveying adjustment,the new algorithm has three advantages:Firstly,this algorithm regards the generalized inverse matrix as a linear operator in the vector space and studies the generalized inverse matrix by studying the mapping law of this linear operator’s corresponding to the basic vector in the vector space.This is a new research perspective in the field of generalized inverse matrix algorithm.Secondly,the main part of this method only involves the calculations of the bases for the three linear space R（A^T）,N（A）and N（A^T）,which avoids a large number of redundant matrix algebra operations and provides a simple form,making it easy to implement computer programs.Thirdly,numerical experiments are implemented on the three most commonly used kinds of generalized inversesA_l（P）^-,A_m（PX）^-andA_PPX⁺.Results show that the proposed algorithm has more accuracy than the algorithm used in the existing surveying adjustment literature does.（3）For the four basic adjustment models of elements,conditions,elements with condition,conditions with parameter and generalized adjustment model,new generalized inverse solution methods for them are given respectively in this thesis,that is,using generalized inverses based on{1}-Inverse to deduce and express their adjustment and precision formula.All the generalized inverses used in the formula are calculated according to the new algorithms in this thesis and the consistency between these new generalized inverse methods and existing methods is also theoretically proved.In addition,the case that the coefficient matrix is row rank deficient in the conditional adjustment is also discussed,and the adjustment formula is given.（4）For the rank deficient free network adjustment,this thesis presents four new generalized inverse solutions.Each solution is based on different linear space theories and uses different types of generalized inverses based on{1}-Inverse to derive different parameter estimation formulas and precision evaluation formulas.The generalized inverses involved in these four solutions are calculated according to the new algorithms raised in this thesis and theoretically prove the consistency between these new generalized inverse methods and existing methods.（5）The algorithm analysis is integrated into the surveying adjustment,that is,the computational complexities of the new generalized inverse solutions for these five adjustment models given in this thesis are analyzed theoretically.The computational efficiency and accuracy of these new solutions are also further tested by numerical experiments.（6）For the five adjustment models of elements,conditions,elements with condition,conditions with parameter and free networks,this thesis compares the efficiency and accuracy of the new generalized inverse algorithm and the existing ones by computational complexity analysis and numerical experiment simulation.The results show that for these five adjustment models,the existing algorithms are more efficient than the new generalized inverse ones proposed in this thesis,while the new generalized inverse algorithms in this thesis have higher calculation accuracy.