Non-linear block least-squares adjustment for a large number of observations

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Abstract
In this contribution two algorithms are developed to solve non-linear system of equations which can contain a large number of measurements. These algorithms are based on nonlinear block least-squares (BLS). Although block least squares was investigated by some researchers, the non-linear case was not examined by now. The first algorithm is proposed to solve a special case of non-linear problems that do not require linearization. Such an algorithm can be called total block least-squares. The second algorithm is based on linearization within a general nonlinear mixed model using a new notation which is in agreement with the rigorous linearization presented by Pope. Both of these algorithms can handle constraints on the parameters. By use of these algorithms, big data processing is feasible with inexpensive computers. Furthermore, expensive processors can solve systems with a large number of equations faster. Two case studies with more than 120,000 equations show that fast and accurate computations are possible by applying these algorithms without any loss of accuracy.

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