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<p>Two three-dimensional 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 equation-solving problem by a 3-D 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 3-D 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 3-D structured networks are proved to converge exponentially fast to the correct solutions. Simulations were performed to show the detailed convergence behaviors of the 3-D structured networks.</p>
three dimensional structured networks; matrix equation solving; linear equations; Lyapunov equation; equation-solving problem; pattern array; linear neurons; training algorithms; Lyapunov methods; matrix algebra; neural nets.

J. Mendel and L. Wang, "Three-Dimensional Structured Networks for Matrix Equation Solving," in IEEE Transactions on Computers, vol. 40, no. , pp. 1337-1346, 1991.
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