TY - JOUR

T1 - 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

AU - Liu, Chein Shan

AU - Atluri, Satya N.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Future cone

KW - Globally Optimal Iterative Algorithm (GOIA)

KW - Nonlinear algebraic equations

KW - Optimal Iterative Algorithm (OIA)

UR - http://www.scopus.com/inward/record.url?scp=84862702621&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84862702621

VL - 84

SP - 575

EP - 601

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 6

ER -