Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem |
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| Authors: | A Dermanis |
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| Institution: | (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: dermanis@topo.auth.gr, GR |
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| 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 x
0-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 x
0-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.
Received: 11 April 1996 / Accepted: 19 April 1997 |
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| Keywords: | , Generalised inverse,Nonlinearity,Pseudoinverse,Inverse problems,Datum problem |
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