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Diakoptics as a general approach in engineering

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Abstract

A method is described for reducing the solution of a matrix equation of linear equations, Mz = d, to that of a number of simpler equations. The method is one mathematical version of diakoptics, and can be used to solve engineering problems involving linear equations without specialization to each individual application. The method has the usual advantages of diakoptics: different parts of the problem can be solved on different computers (distributed computation), decreased computer storage requirements in fast storage, applicability to some types of parallel processors and decreased computation time in some circumstances. Consideration is given to reducing M to specific forms including block triangular, banded and the usual block diagonal. Consideration is also given to the case where M is not square and a new application is given to solving least-squares problems.

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References

  1. P.W. Aitchison: Diakoptics or tearing — a mathematical approach, Quarterly of Applied Mathematics 41 (1983) 265–272.

    Google Scholar 

  2. H. Asai and M. Tanaka: Diakoptics: multi-level tearing method and estimation of the computation time, Proceedings of International Symposium on Circuits and Systems, 1984, Vol. 1, IEEE, New York (1984) 364–367.

    Google Scholar 

  3. Adi Ben-Israel and T.N.E. Greville: Generalized Inverses: Theory and Applications, Wiley, New York (1974), reprinted by Krieger (1980).

    Google Scholar 

  4. M.V. Bhat and H.K. Kesaven: Diakoptic equations and sparsity, IEEE PES Summer Meeting, Anaheim (1974); IEEE (1974) paper C74.384-4.

  5. M.D. Bradshaw: An electromagnetic analog to the method of diakoptics (Kron's ‘tearing technique’) of circuit analysis, 23rd. Midwest Symposium on Circuits and Systems, Toledo (1980) 562–566.

  6. A. Brameller, M.N. John and M.R. Scott: Practical Diakoptics for Electrical Networks, Chapman and Hall, London (1969).

    Google Scholar 

  7. A. Brameller and K.L. Lo: The application of diakoptics and the escalator method to the solution of very large eigenvalue problems, International journal for Numerical Methods in Engineering 2 (1970) 535–549.

    Google Scholar 

  8. H.F. Bückner: Numerical methods for integral equations; in: Survey of Numerical Analysis, McGraw-Hill, New York (1962).

    Google Scholar 

  9. J.R. Bunch and D.J. Rose: Partitioning, tearing and modification of sparse linear systems, Journal of Mathematical Analysis and Applications 48 (1974) 574–593.

    Google Scholar 

  10. G. Cafaro, P. Pugliese and F. Vacca: Parallel solution of turn networks, International Journal of Electrical Power and Energy Systems 6 (1984) 131–138.

    Google Scholar 

  11. C.F. Chen and K.U. Wang: Diakoptics for eigenvalues of large scale networks, 23rd Midwest Symposium on Circuits and Systems, Toledo (1980); North Holland, California (1980) 572–574.

    Google Scholar 

  12. L.O. Chua and L. Chen: Diakoptic and generalized hybrid analysis, IEEE Transactions on Circuits and Systems 23 (1976) 694–705.

    Google Scholar 

  13. J. Constantinescu: Study of the transient processes in large-scale power systems, Revue Roumaine des Sciences Techniques (Serie d'Electrotechnique et Energetique) 27 (1982) 211–227.

    Google Scholar 

  14. P. Cristea and M. Iordache: A diakoptic analysis method, IEEE International Symposium on Circuits and Systems (1983), IEEE, New York (1983) 633–636.

    Google Scholar 

  15. I.S. Duff and J.K. Reid: An implementation of Tarjan's algorithm for the block triangularization of a matrix, ACM Transactions on Mathematical Software 4 (1978) 137–147 and 189–192.

    Google Scholar 

  16. M. El-Marsafawy, R.W. Menzies and R.M. Mathur: New diakoptic technique for load-flow solution of very large power-systems using the bus admittance matrix, Proc. Institution of Electrical Engineers 126 (1979) 1301–1302.

    Google Scholar 

  17. T. Tukao and E. Teratsuji: An extension of diakoptics and its application to singular decomposition problem of networks, Electrical Engineering in Japan 102 (1) (1982) 141–148.

    Google Scholar 

  18. A. George and J.W. Liu: Computer Solution of Large Sparse Positive Definite Systems, Prentice Hall, New Jersey (1981) Ch. 4

    Google Scholar 

  19. N.E. Gibbs, W.G. Poole and P.K. Stockmeyer: An algorithm for reducing the band width and profile of a sparse matrix, SIAM Journal of Numerical Analysis 13 (1976) 236–250.

    Google Scholar 

  20. G. Goubao, N.N. Puri and F.K. Schwering: Diakoptic theory for multi-element antennas, IEEE Transactions on Antennas and Propagation 30 (1982) 15–26.

    Google Scholar 

  21. I.N. Hajj: Diakoptics and block elimination, IEEE 1978 Summer Power Meeting at Los Angeles, paper A78, 564–7.

  22. I.N. Hajj: Sparsity considerations in network solution by tearing, IEEE Transactions on Circuits and Systems 27 (1980) 357–366.

    Google Scholar 

  23. H.H. Happ: The Application of diakoptics to the solution of power system problems, in: Electrical Power Problems, The Mathematical Challenge, SIAM (1980) 69–103.

  24. H.H. Happ: Piecewise Methods and Applications to Power Systems, John Wiley, Chichester (1980).

    Google Scholar 

  25. H.V. Henderson and S.R. Searle: On deriving the inverse of a sum of matrices, SIAM Review 23 (1981) 53–60.

    Google Scholar 

  26. P. Henrici: Discrete Variable Methods in Ordinary Differential Equations, John Wiley, New York (1962).

    Google Scholar 

  27. P.B. Johns and K. Akhtarzad: Time domain approximations in the solution of fields by time domain diakoptics, International Journal for Numerical Methods in Engineering 18 (1982) 1361–1373.

    Google Scholar 

  28. H.K. Kesaven and J. Dueckman: Multi-terminal representations and diakoptics, Journal of the Franklin Institute 313 (1982) 337–352.

    Google Scholar 

  29. A. Klos: What is diakoptics? Electrical Power and Energy Systems 4 (1982) 192–195.

    Google Scholar 

  30. I.N. Kotarova and O.Yu. Shamaeva: A parallel diakoptics method of solving complex problems on distributed computational systems, Cybernetics 15 (1979) 131–141.

    Google Scholar 

  31. G. Kron: Diakoptics — piecewise solution of large scale systems, The Electrical Journal, in 20 parts from 158 (1957) 1673–1677 to 162 (1959) 431–436.

    Google Scholar 

  32. B. Majundar and T.N. Saha: Diakoptic approach to symmetrical short circuit analysis, Journal of the Institution of Engineers (India), part EL, Electrical Engineering Division 61 (1980) 1–4.

    Google Scholar 

  33. S.Y. Mansour, D.H. Kelly and D.O. Koval: Diakoptical active-reactive dispatch of generation, Proceedings of the 1984 IEEE International Symposium on Circuits and Systems, Vol. 1, IEEE, New York (1984) 136–139.

    Google Scholar 

  34. O.W. Marcus, K. Reiss, B.X. Weis and Z.P. Tomaszewski: Basic considerations on an adaptive iteration algorithm for large scale network analysis, Proceedings of the 1983 IEEE Symposium on Circuits and Systems, Vol. 1, IEEE, New York (1983) 230–233.

    Google Scholar 

  35. J. Myslik: Diakoptical analysis of the networks considering the changes of branch impedances, Electrotech. Obz. (Czechoslovakia) 73 (1984) 196–2000.

    Google Scholar 

  36. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling: Numerical Recipes: the Art of Scientific Computing, Cambridge University Press, Cambridge (1986).

    Google Scholar 

  37. N.B. Rabatt and H.Y. Hsieh: A latent macromodular approach to large-scale sparse networks, IEEE Transactions on Circuits and Systems 23 (1976) 745–752.

    Google Scholar 

  38. J.K. Reid: A survey of sparse matrix computation, in: Electrical Power Problems: the Mathematical Challenge, SIAM (1980) 41–68.

  39. A. Sangiovanni-Vincentelli, L. Chen and L.O. Chua: A new tearing approach-node tearing nodal analysis, Proceedings of 1977 IEEE International Symposium on Circuits and Systems, IEEE, New York (1977) 143–147.

    Google Scholar 

  40. D.V. Steward: Partitioning and tearing of systems of equations, SIAM J. Numerical Analysis, Series B, 2 (1965) 345–365.

    Google Scholar 

  41. P.K.U. Wang and C.F. Chen: Sensitivity calculation and network optimization through decomposition, Proceedings of 1983 IEEE Symposium on Circuits and Systems, Vol. 3, IEEE, New York, (1983) 1034–1037.

    Google Scholar 

  42. F.F. Wu: Solution of large-scale networks by tearing, IEEE Transactions on Circuits and Systems 23 (1976) 706–713.

    Google Scholar 

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  1. Department of Applied Mathematics, University of Manitoba, R3T 2N2, Winnipeg, Manitoba, Canada

    P. W. Aitchison

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Aitchison, P.W. Diakoptics as a general approach in engineering. J Eng Math 21, 47–58 (1987). https://doi.org/10.1007/BF00127693

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  • DOI: https://doi.org/10.1007/BF00127693

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