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Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem

Authors
  • Dermanis, A.1
  • 1 Department of Geodesy and Surveying, The Aristotle University of Thessaloniki, University Box 503, GR-54006 Thessaloniki, Greece Phone: +30 31 996111; fax: +30 31 996137; email: [email protected]th.gr, GR
Type
Published Article
Journal
Journal of Geodesy
Publisher
Springer-Verlag
Publication Date
Feb 01, 1998
Volume
72
Issue
2
Pages
71–100
Identifiers
DOI: 10.1007/s001900050150
Source
Springer Nature
Keywords
License
Yellow

Abstract

Motivated by the existing theory of the geometric characteristics of linear generalized inverses of linear mappings, an attempt is made to establish a corresponding mathematical theory for nonlinear generalized inverses of nonlinear mappings in finite- dimensional spaces. The theory relies on the concept of fiberings consisting of disjoint manifolds (fibers) in which the domain and range spaces of the mappings are partitioned. Fiberings replace the quotient spaces generated by some characteristic subspaces in the linear case. In addition to the simple generalized inverse, the minimum-distance and the x0-nearest generalized inverse are introduced and characterized, in analogy with the least-squares and the minimum-norm generalized inverses of the linear case. The theory is specialized to the geodetic mapping from network coordinates to observables and the nonlinear transformations (Baarda's S-transformations) between different solutions are defined with the help of transformation parameters obtained from the solution of nonlinear equations. In particular, the transformations from any solution to an x0-nearest solution (corresponding to Meissl's inner solution) are given for two- and three-dimensional networks for both the similarity and the rigid transformation case. Finally the nonlinear theory is specialized to the linear case with the help of the singular-value decomposition and algebraic expressions with specific geometric meaning are given for all possible types of generalized inverses.

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