Bulletin géodésique

, Volume 51, Issue 3, pp 177–202 | Cite as

A few basic principles and techniques of array algebra

  • G. Blaha
Article

Abstract

This paper is intended to demonstrate the usefulness of array algebra techniques in certain multilinear least squares problems. A typical restriction of array algebra is the need for a gridded observational structure; however, the grid does not have to be uniform and in general is not limited to any particular coordinate system nor to two- or three-dimensional spaces. Another restriction comes to light when dealing with weighted multilinear least squares adjustments. The a—priori variance-covariance matrix cannot be completely arbitrary but must be expressible in terms of certain matrix products. There exist various practical ways (not discussed herein) to bridge these restrictions. The reward for using the array algebra technique when it is appropriate lies in the great computational savings.

From the theoretical point of view, the backbone of most derivations are the “R-matrix multiplications” and a simple tool, demonstrated herein, called “fundamental transformation”. It follows that the least squares solution of “array observation equations” does not have to be sought by some new and complex mathematical means. The fundamental transformation allows such an adjustment problem to be rewritten in a conventional (monolinear) form; the familiar least squares solution is then written down and transformed back to the array form using the same tool. The statistical properties of the results (e.g. minimum variance) are known from the conventional approach and do not have to be rederived in the array case.

Keywords

Scalar Multiplication Brute Force Parallel Processor Weighted Case Array Formulation 

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References

  1. G. BLAHA:Inner Adjustment Constraints with Emphasis on Range Observations. Department of Geodetic Science, Report No. 148, The Ohio State University, Columbus, 1971.Google Scholar
  2. U.A. RAUHALA: “Array Algebra with Applications in Photogrammetry and Geodesy”. Paper published inFotogrammetriska Meddelanden V1: 6, Department of Photogrammetry, Royal Institute of Technology, Stockholm, Sweden, 1974.Google Scholar
  3. U.A. RAUHALA: “A Review of Array Algebra”. Paper published inFotogrammetriska Meddelanden, 2: 38, Department of Photogrammetry, Royal Institute of Technology, Stockholm, Sweden, 1976.Google Scholar
  4. U.A. RAUHALA and S. GERIG:Digital Terrain Data Compaction Using Array Algebra. Report of DBA Systems, Inc; performed for USA-ETL, Fort Belvoir, Virginia, under Contract No. DAAG53-76-C-0127, November, 1976.Google Scholar

Copyright information

© Bureau Central de L’Association Internationale de Géodésie 1977

Authors and Affiliations

  • G. Blaha
    • 1
  1. 1.DBA Systems, Inc.Melbourne

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