doi:10.3970/cmes.2012.084.575

Source | CMES: Computer Modeling in Engineering & Sciences, Vol. 84, No. 6, pp. 575-602, 2012 |

Download | Full length paper in PDF format. Size = 392,700 bytes |

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 a_{c} in the best descent vector u=a_{c} F+B^{\unhbox \voidb@x 89.5 T}F, 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=a_{c}F+B^{\unhbox \voidb@x 89.5 T}F; 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. |