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Calculation of loop inverses / Urho Rauhala (1976)

Public contenu dans [Two papers by Techn Dr Urho Rauhala] / Urho Rauhala (1976) Titre : Calculation of loop inverses Type de document : Article/Communication Auteurs : Urho Rauhala, Auteur Editeur : Stockholm : Royal Institute of Technology Année de publication : 1976 Collection : Fotogrammetriska meddelanden, ISSN 0071-8068 num. 2-38 Conférence : ASP 1975, Spring Convention Washington DC Washington DC - Etats-Unis Importance : pp 40 - 78 Format : 21 x 30 cm Note générale : bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Analyse numérique [Termes IGN] algèbre linéaire [Termes IGN] calcul matriciel [Termes IGN] factorisation de Cholesky [Termes IGN] inverse généralisé [Termes IGN] tenseur Résumé : (auteur) The concept of loop inverses, originates from the interpolation technique using assumed grid observations as unknowns in the "calculus of matrix array', Rauhala (1972, 1972b). The theory of loop inverses was developed in the "array algebra" of Rauhala (1974) resulting in a unified and generalized theory of generalized matrix inverses. Array calculus use of Einstein's summation convention for array indices and the of so-called R-vectors and R-matrices in a more general way than is possible in the conventional vector, matrix end tensor algebras. Extremely large multidimensionaI and multisingular equation systems for grid observations may be effectively solved using array algebra and modern computers. Herein a brief presentation of some special loop inverses concerning mainly the combinations of 1- and m- inverses or the well known full rank least squares inverse of the element adjustment and the minimum norm -inverse of the condition adjustment is made. As an example the lm- inverses derived in more details and is seen to belong to the category of constrained inverse of Rao and Mitra (1971) but not necessarily to fill the definition of Bjerhammar (1955) for a g-inverse. As a special case the lm-inverse creates the Moore (1920) - Penrose (1955) pseudoinverse. The general loop inverses fill only the more general (weaker) condition YaY = Y instead of the more restricted condition AyA = A of the g—inverse. Some computational aspects and examples of applications of the le-inverse and pseudoinverse are given. An algorithm of "Array Cholesky" is deveIoped which a) allows singularities b) handles singular rectangular matrices as elements c) works directly on the observation equations without the intermediate stage of the building normal equations d) preserves the sparseness of the “not built” normal equations e) needs no intermediate inversion of the triangular hyper matrix in calculation of the solution and the weight coefficient matrix f) is suited to the solution of ill-conditioned systems by a priori choice and reordering of the most significant parameters g) is suited for array solutions. Outlines are drawn for the application of this method in the general simultaneous adjustment of photogrammetric and geodetic observations. Numéro de notice : C1976-003 Affiliation des auteurs : non IGN Thématique : MATHEMATIQUE Nature : Communication Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=92981