Diagramatic approach to solve least-squares adjustment and collocation problems

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The standard method of obtaining the solution to the least-squares adjustment problems lacks objectivity and can not readily be interpreted geometrically. Thus far, one approach has been made to interpret the adjustment solution geometrically, using Hilbert space technique. This thesis is still another effort in this respect; it illustrates the important advantage of considering the parallelism between the concept of a metric tensor and a covariance matrix. By splitting the linear mathematical models in the least-squares adjustment (parametric, conditional and combined models) into dual covariant and contravariant spaces, this method produces diagrams for the least-squares adjustment and collocation, from which the necessary equations may be obtained. Theories and equations from functional analysis and tensor calculus, necessary for the analysis of this approach, are described. Also the standard method used in the least-squares adjustment and the least-squares collocation are reviewed. In addition, the geometrical interpretation of a singular weight matrix is presented.