Abstract
Many problems in physical science involve the estimation of a number of unknown parameters which bear a linear (or linearized) relationship to a set of experimental data. The data may be contaminated by (systematic or random) errors, insufficient to determine the unknowns, redundant, or all of the above and consequently, questions as existence, uniqueness, stability, approximation and the physical description of the set of solutions are all of interest.
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References
Books in the mathematical literature which deal with the theory of generalized inverses are:
Ben-Israel, A. and T.N.E. Greville (1974). Generalized Inverses: Theory and Applications. J. Wiley, New York.
Bouillon, T.L. and P.L. Odell (1971). Generalized Inverse Matrices. Wiley-Interscience, New York.
Rao, C.R. and S.K. Mitra (1971). Generalized Inverse of Matrices and its Applications. J. Wiley, New York. The book by and mitra is probably the most cited references in the geodetic literature dealing with generalized inverses.
Books written by and for geodesists which treat the theory of generalized inverses and give geodetic application are:
Bjerhammar, A. (1973). Theory of Errors and Generalized Matrix Inverses. Elsevier, Amsterdam. Bjerhammar was one of the first who started a systematic study of the problem of inverting of matrices of arbitrary order and rank. (see e.g. his paper [Bjerhammar, 1951].).
Grafarend, E., H. Heister, R. Keim, H. Kropff, and B. Schaffrin (1979). Optimierung Geodätischer Mess Operationen. Herbert Wichmann Verlag Karlsruhe. Band II.
Koch, K.R. (1980). Parameterschätzung und Hypothesentests in linearen Modellen. Dümmler.
Meissl, P. (1982). Least Squares Adjustment: A Modern Approach. Mitteilungen der Geodätischen Institute der Technischen Universität Graz. Folge 43.
Where scientific papers are concerned, there is an overwhelming list of geodetic papers which deal with the theory of generalized inverses and s-transformations. Here are some typical examples:
Baarda, W. (1973). S-transformations and Criterion Matrices, Netherlands Geodetic Commission, Publications on Geodesy, New Series Vol. 5, No. 1, 1973 Delft. Baarda was the first who made a systematic study of the consequences of datum definitions on the precision description of geodetic networks. (For an application of the theory of s-transformations to the problem of precision testing in geodetic networks, see also my lecture notes [Teunissen, Quality Control in Geodetic Networks]).
Bjerhammar, A. (1951). Rectangular Reciprocal Matrices, with Special Reference to Geodetic Calculations, No. 20, 188–220.
Blaha, G. (1971). Inner Adjustment Constraints with Emphasis on Range Observations. Report No. 148, Department of Geodetic Science of the Ohio State University, Columbus.
Grafarend, E. and B. Schaffrin (1974). Unbiased Free Net Adjustment, Survey Review 22 (1974), 200–218.
Grafarend, E. and B. Schaffrin (1976). Equivalence of Estimable Quantities and Invariants in Geodetic Networks, zfv 101, S. 485–491.
Meissl, P. (1962). Die Innere Genauigkeit eines Punkthaufens, Öz Vermessungswesen 50 (1962), 159–165, 186–194.
Mierlo, J. van (1979). Free Network Adjustment and S-transformations, DGK B, Nr. 252, S. 41–54. München 1980.
Mittermayer, E. (1971). Zur Ausgleichung Freier Netze. Zfv 97, S. 481–489.
Pelzer, H. (1974). Zur Behandlung Singulärer Ausgleichungsproblemen, Zfv 99, S. 181–194.
Pope, A. (1971). Transformation of Covariance Matrices Due to Changes in Minimal Control, Paper pres. American Geophysical Union, San Francisco 1971
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Teunissen, P. (1985). Zero Order Design: Generalized Inverses, Adjustment, the Datum Problem and S-Transformations. In: Grafarend, E.W., Sansò, F. (eds) Optimization and Design of Geodetic Networks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-70659-2_3
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