A Globally Optimal Iterative Algorithm Using the Best Descent Vector \mathaccentV dot05Fx=l[ac F+BTF], with the Critical Value ac, for Solving a System of Nonlinear Algebraic Equations F(x)=0
Chein-Shan Liu; Satya N. Atluri

doi:10.3970/cmes.2012.084.575
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 84, No. 6, pp. 575-602, 2012
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Keywords Nonlinear algebraic equations, Future cone, Optimal Iterative Algorithm (OIA), Globally Optimal Iterative Algorithm (GOIA)
Abstract An iterative algorithm based on the concept of best descent vector u in \mathaccentV dot05Fx=lu is proposed to solve a system of nonlinear algebraic equations (NAEs): F(x)=0. In terms of the residual vector F and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm evolves. A new method to approximate the best descent vector is developed, and we find a critical value of the weighting parameter ac in the best descent vector u=ac F+B\unhbox \voidb@x 89.5 TF, where B=F/x is the Jacobian matrix. We can prove that such an algorithm leads to the largest convergence rate with the descent vector given by u=acF+B\unhbox \voidb@x 89.5 TF; hence we label the present algorithm as a globally optimal iterative algorithm (GOIA). Some numerical examples are used to validate the performance of the GOIA; a very fast convergence rate in finding the solution is observed.
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