Array algebra expansion of matrix and tensor calculus: Part 1

SIAM Journal on Matrix Analysis and Applications

Volumn 24, Issue 2, 2003, Pages 490-508

  Rauhala, Urho A ^a  

^a BAE SYSTEMS   (United States)

Author keywords

Array relaxation; Decentered normals; Fast transforms; Loop inverse estimation; Multigrid normals; Multilinear array algebra; Nonlinear array polynomials; Nonlinear tensor methods; Q surface; Tensor product

Indexed keywords

EID: 0037256037     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S089547980240683X     Document Type: Article

References (33)

1. T. M. APOSTOL, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, MA, 1957.
2. W. BAARDA, S-Transformations and Criterion Matrices, Netherlands Geodetic Commission 5, Delft University of Technology, Delft, The Netherlands, 1973.
3. A. BJERHAMMAR, Rectangular reciprocal matrices with special reference to geodetic calculations, Bull. Géodésique, 1951, pp. 188-220.
4. G. BLAHA, A few basic principles and techniques of array algebra, Bull. Gédésique, 51 (1977), pp. 177-202.
5. G. BLAHA, Non-iterative approach to nonlinear least-squares adjustment, Manuscripta Geodaetica, 19 (1994), pp. 199-212.
6. H. EBNER AND P. REISS, Height interpolation by the method of finite elements, in Proceedings of the DTM Symposium, (St. Louis, MO, 1978), ASPRS, VA, 1978, pp. 241-254.
7. J. COOLEY AND J. TUKEY, An algorithm for machine computation of complex Fourier series, Math. Comp., 19 (1965), pp. 297-301.
8. L. DE LATHAUWER, B. DE MOOR, AND J. VANDEWILLE, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278.
9. D. W. FAUSETT AND C. T. FULTON, Large least squares problems involving Kronecker products, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 219-227.
10. I. J. GOOD, The interaction algorithm and practical Fourier series, J. Roy. Statist. Soc. Ser. B, 20 (1958), pp. 361-372.
11. E. GRAFAREND AND B. SCHAFFRIN, Equivalence of Estimable Quantities and Invariants in Geodetic Networks, Z. Vermessungwesen, 11 (1976).
12. M. GU, New fast algorithms for structured linear least squares problems, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 244-269.
13. V. KRATKY, Grid-modified polynomial transformation of satellite imagery, Remote Sensing Environ, 5 (1976), pp. 67-74.
14. B. NOBLE, Methods for computing the Moore-Penrose generalized inverse and related matters, Generalized Inverses and Applications, M. Z. Nashed, ed., Academic Press, New York, 1976. pp. 245-301.
15. R. PENROSE, On best approximation solutions of linear matrix equations, Proc. Cambridge Philos. Soc., 52 (1956), pp. 17-19.
16. A. POPE, Modern trends in adjustment calculus, in Proceedings of the International Symposium of NA Geodetic Networks, New Brunswick, NJ, 1974.
17. C. R. RAO AND S. K. MITRA, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971.
18. U. A. RAUHALA, Calculus of Matrix Arrays and General Polynomial and Harmonic Interpolation by Least Squares with New Solutions in Photogrammetry and Geodesy, Fot. Medd. VI:4, Department of Photogrammetry, Royal Institute of Technology, Stockholm, 1972.
19. U. A. RAUHALA, Array Algebra with Applications in Photogrammetry and Geodesy, Fot. Medd. VI:6, Department of Photogrammetry, Royal Institute of Technology, Stockholm, 1974.
20. U. A. RAUHALA, Array algebra as general base of fast transforms, in Proceedings of the Symposium on Image Processing - Interaction with Photogrammetry and Remote Sensing, Mitteilungen Der Geodaetischen Inst. Der T U Graz, Folge 29, 1977, pp. 175-188.
21. U. A. RAUHALA, Intuitive derivation of loop inverses and array algebra, Bull. Géodésique, 53 (1979), pp. 317-342.
22. U. A. RAUHALA, Introduction to array algebra, Photogrammetric Engrg. Remote Sensing, 46 (1980), pp. 177-192.
23. U. A. RAUHALA, Note on general linear estimators and matrix inverses, Manuscripta Geodaetica, 6 (1981), pp. 375-386; technical report 80-015, Geodetic Services, Inc., Melbourne, FL.
24. U. A. RAUHALA, Compiler Positioning of Array Algebra Technology, International Society of Photogrammetry and Remote Sensing, Vol. 26-3/3, Comm. III Symposium, Rovaniemi, 1986, pp. 173-198.
25. U. A. RAUHALA, General theory of array algebra in nonlinear least squares and robust estimation, ASPRS Spring Convention, Denver, 1990.
26. U. A. RAUHALA, Nonlinear Array Algebra in Digital Photogrammetry, International Society of Photogrammetry and Remote Sensing, Vol. 29 B2 II (1992), pp. 95-102.
27. U. A. RAUHALA, D. DAVIS, AND K. BAKER, Automated DTM validation and progressive sampling algorithm of finite element array relaxation, Photogrammetric Engrg. Remote Sensing, 4 (1989), pp. 449-465.
28. P. A. REGALIA AND S. K. MITRA, Kronecker products, unitary matrices and signal processing applications, SIAM Rev., 31 (1989), pp. 586-613.
29. P. ROZSA AND I. TOTH, Eine direkte Methode zur Losung der Poissonschen Differentialgleichung mit Hilfe des 9- Punkte-Verfahrens, ZAMM Z. Angew. Math. Mech., 50 (1970), pp. 713-720.
30. R. B. SCHNABEL AND P. D. FRANK. Tensor methods for nonlinear equations, SIAM J. Numer. Anal., 21 (1984), pp. 815-843.
31. R. A. SNAY, Applicability of array algebra, Reviews of Geophysics and Space Physics, 16 (1978), pp. 459-464.
32. M. SUZUKI AND K. SHIMIZU, Analysis of distributed systems by array algebra, International J. Systems Sci., 21 (1990). pp. 129-155.
33. C. VAN LOAN AND N. PITSIANIS, Approximation with, Kronecker products, in Linear Algebra for Large Scale and Real Time Applications, M. S. Moonen and G. H. Golub, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, pp. 293-314.