TY - EJOU
AU - Liu, Chein-Shan
AU - Atluri, Satya N.
TI - A Globally Optimal Iterative Algorithm Using the Best Descent Vector x^{·} = λ[α_{c}F + B^{T}F], with the Critical Value α_{c}, for Solving a System of Nonlinear Algebraic Equations F(x) = 0
T2 - Computer Modeling in Engineering \& Sciences
PY - 2012
VL - 84
IS - 6
SN - 1526-1506
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 α_{c} in the best descent vector **u = α**_{c}F + B^{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 = α**_{c}F + B^{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.
KW - Nonlinear algebraic equations
KW - Future cone
KW - Optimal Iterative Algorithm (OIA)
KW - Globally Optimal Iterative Algorithm (GOIA)
DO - 10.3970/cmes.2012.084.575