A globally optimal iterative algorithm using the best descent vector x =λ [αcF+BTF], with the critical value αc, for solving a system of nonlinear algebraic equations F(x) = 0

Chein Shan Liu, Satya N. Atluri

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

An iterative algorithm based on the concept of best descent vector u in x =λu 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 = αcF+BTF, 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 =αcF+BTF; 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.

Original languageEnglish
Pages (from-to)575-601
Number of pages27
JournalCMES - Computer Modeling in Engineering and Sciences
Volume84
Issue number6
StatePublished - 2012

Keywords

  • Future cone
  • Globally Optimal Iterative Algorithm (GOIA)
  • Nonlinear algebraic equations
  • Optimal Iterative Algorithm (OIA)

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