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A closed-form of Newton method for solving over-determined pseudo-distance equations

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Abstract

The Newton method has been widely used for solving nonlinear least-squares problem. In geodetic adjustment, one would prefer to use the Gauss–Newton method because of the parallel with linear least-squares problem. However, it is proved in theory as well as in practice that the Gauss–Newton method has slow convergence rate and low success rate. In this paper, the over-determined pseudo-distance equations are solved by nonlinear methods. At first, the convergence of decent methods is discussed after introducing the conditional equation of nonlinear least squares. Then, a compacted form of the Hessian matrix from the second partial derivates of the pseudo-distance equations is given, and a closed-form of Newton method is presented using the compacted Hessian matrix to save the computation and storage required by Newton method. At last, some numerical examples to investigate the convergence and success rate of the proposed method are designed and performed. The performance of the closed-form of Newton method is compared with the Gauss–Newton method as well as the regularization method. The results show that the closed-form of Newton method has good performances even for dealing with ill-posed problems while a great amount of computation is saved.

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Acknowledgments

We would like to thank the editors and the referees of this paper for their helpful comments. This work is partly supported by National Science Foundation of China (Grant No. 41020144004, 41104018), National High-tech R&D Program (grant No. 2009AA121405,2012BAB16B01) and GFZX0301040308-06.

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Authors and Affiliations

  1. Chinese Academy of Surveying and Mapping, Beijing , 100830, China

    Shuqiang Xue & Yamin Dang

  2. National Key Laboratory for Geo-information Engineering, Xi’an Research Institute of Surveying and Mapping, Xi’an , 710054, China

    Yuanxi Yang

  3. School of Geological and Surveying Engineering, Chang’an University, Yanta Road, Xi’an , 710054, China

    Shuqiang Xue

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  1. Shuqiang Xue
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  2. Yuanxi Yang
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  3. Yamin Dang
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Correspondence to Shuqiang Xue.

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Xue, S., Yang, Y. & Dang, Y. A closed-form of Newton method for solving over-determined pseudo-distance equations. J Geod 88, 441–448 (2014). https://doi.org/10.1007/s00190-014-0695-y

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