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L.X. Wang, J.M. Mendel, "ThreeDimensional Structured Networks for Matrix Equation Solving," IEEE Transactions on Computers, vol. 40, no. 12, pp. 13371346, December, 1991.  
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@article{ 10.1109/12.106219, author = {L.X. Wang and J.M. Mendel}, title = {ThreeDimensional Structured Networks for Matrix Equation Solving}, journal ={IEEE Transactions on Computers}, volume = {40}, number = {12}, issn = {00189340}, year = {1991}, pages = {13371346}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.106219}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  ThreeDimensional Structured Networks for Matrix Equation Solving IS  12 SN  00189340 SP1337 EP1346 EPD  13371346 A1  L.X. Wang, A1  J.M. Mendel, PY  1991 KW  three dimensional structured networks; matrix equation solving; linear equations; Lyapunov equation; equationsolving problem; pattern array; linear neurons; training algorithms; Lyapunov methods; matrix algebra; neural nets. VL  40 JA  IEEE Transactions on Computers ER   
Two threedimensional structured networks are developed for solving linear equations and the Lyapunov equation. The basic idea of the structured network approaches is to first represent a given equationsolving problem by a 3D structured network so that if the network matches a desired pattern array, the weights of the linear neurons give the solution to the problem: then, train the 3D structured network to match the desired pattern array using some training algorithms; and finally, obtain the solution to the specific problem from the converged weights of the network. The training algorithms for the two 3D structured networks are proved to converge exponentially fast to the correct solutions. Simulations were performed to show the detailed convergence behaviors of the 3D structured networks.
[1] D. P. Bertsekas and J. N. Tsitsiklis,Parallel and Distributed Computations. Englewood Cliffs, NJ: PrenticeHall, 1989.
[2] D. Heller, "A survey of parallel algorithms in numerical linear algebra,"SIAM Rev., vol. 20, pp. 740777, 1978.
[3] A. H. Sameh and D. J. Kuck, "On stable parallel linear solver,"J. ACM, vol. 25, pp. 8191, 1978.
[4] S. L. Johnson, "Solving tridiagonal systems on ensemble architectures,"SIAM J. Sci. Stat. Comput., vol. 8, pp. 345392, 1987.
[5] R. E. Lord, J. S. Kowalik, and S. P. Kumar, "Solving linear algebraic expressions on an MIMO computer,"J. ACM, vol. 30, pp. 103117, 1983.
[6] A. Bojanczyk, R. P. Brent, and H. T. Kung, "Numerically stable solution of linear equations using meshconnected processors,"SIAM J. Sci. Stat. Comput., vol. 5, pp. 95104, 1984.
[7] S.Y. Kung,VLSI Array Processors, Prentice Hall, Englewood Cliffs, N.J. 1988.
[8] H. T. Kung, B. Sproul, and G. Steele,VLSI Systems and Computations. Rockville, MD: Computer Science Press, 1981.
[9] L. Csanky, "Fast parallel matrix inversion algorithms,"SIAM J. Comput., vol. 5, pp. 618623, 1976.
[10] A. Bojanczy, "Complexity of solving linear systems in different models of computation,"J. ACM, vol. 32, pp. 792803, 1984.
[11] S. Lo, B. Phillipe, and A. Sameh, "A multiprocessor algorithm for the symmetric tridiagonal eigenvalue problem,"SIAM J. Sci. Stat. Comput., vol. 8, pp. s155s165, 1987.
[12] J. Dongarra and D. Sorensen, "A fully parallel algorithm for the symmetric eigenvalue problem,"SIAM J. Sci. Stat. Comput., vol. 8, pp. s139s154, 1987.
[13] R. Barlow, D. Evans, and J. Shanehchi, "Parallel multisection applied to the eigenvalue problem,"The Comput. J., vol. 26, pp. 69, 1983.
[14] E. Jessup and D. Sorensen, "A multiprocessor scheme for the singular value decomposition," inParallel Processing for Scientific Computing, 1989, pp. 6166.
[15] E. Jessup, "Parallel solution of the symmetric tridiagonal eigenproblem," Res. Rep. YALEU/DCS/RR728, 1989.
[16] A. Brenneret al., "Panel: How do we make parallel processing a reality?: Bridging the gap between theory and practice," inProc. 5th Int. Parallel Processing Symp., 1991, pp. 648653.
[17] N. Morgan, Ed.,Artificial Neural Networks Electronic Implementations. IEEE Computer Society Press, 1990.
[18] U. Ramacher and U. Ruckert, Eds.,VLSI Design of Neural Networks. Norwell, MA: Kluwer Academic, 1991.
[19] L. X. Wang and J. M. Mendel, "Structured trainable networks for matrix algebra," inProc. 1990 Int. Joint Conf. Neural Networks, vol. 2, 1990, pp. II125II132.
[20] L. X. Wang and J. M. Mendel, "Matrix computations and equation solving using structured networks and training," inProc. IEEE 1990 Conf. Decision and Control, 1990, pp. 17471750.
[21] L. X. Wang and J. M. Mendel, "Parallel structured networks for solving a wide variety of matrix algebra problems,"J. Parallel Distributed Processing, Mar. 1992, to be published.
[22] M. M. Polycarpou and P. A. Ioannou, "Learning and convergence analysis of neuraltype structured networks,"IEEE Trans. Neural Networks, 1991, to be published.
[23] L. X. Wang and J. M. Mendel, "Cumulantbased parameter estimation using structured networks,"IEEE Trans. Neural Networks, vol. 2, no. 1, pp. 7383, 1991.
[24] P. Werbos, "New tools for predictions and analysis in the behavioral science," Ph.D. dissertation, Harvard Univ. Committee on Applied Mathematics, 1974.
[25] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning internal representation by error propagation,"Parallel Distributed Processing: Explorations in the Microstructure of Cognition, vols. 1 and 2. Cambridge, MA: MIT Press, 1986.
[26] D. B. Parker, "Learning logic," Tech. Rep. TR47, M.I.T. Center for Computational Economics and Statistics, 1985.
[27] B. Widrow and S. D. Stearns,Adaptive Signal Processing. Englewood Cliffs, NJ: PrenticeHall, 1985.
[28] D. G. Luenberger, Optimization byVector Space Methods. New York: Wiley, 1969.
[29] J. W. Brewer, "Kronecker products and matrix calculus in system theory,"IEEE Trans. Circuits Syst., vol. CAS25, pp. 772781, 1978.