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This paper presents an MLP-type neural network with some fixed connections and a backpropagation-type training algorithm that identifies the full set of solutions of a complete system of nonlinear algebraic equations with n equations and n unknowns. The proposed structure is based on a backpropagation-type algorithm with bias units in output neurons layer. Its novelty and innovation with respect to similar structures is the use of the hyperbolic tangent output function associated with an interesting feature, the use of adaptive learning rate for the neurons of the second hidden layer, a feature that adds a high degree of flexibility and parameter tuning during the network training stage. The paper presents the theoretical aspects for this approach as well as a set of experimental results that justify the necessity of such an architecture and evaluate its performance. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY We consider the numerical solution of the continuous algebraic Riccati equation A* X + XA − XFX + G = 0, with F = F*, G = G* of low rank and A large and sparse. We develop an algorithm for the low-rank approximation of X by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought-after approximation can be obtained by a low-rank update, in the style of the well known Alternating Direction Implicit (ADI) iteration for the linear equation, from which the new method inherits many algebraic properties. Moreover, we establish new insightful matrix relations with emerging projection-type methods, which will help increase our understanding of this latter class of solution strategies. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]