TY - JOUR

T1 - Iterative solution of a system of Nonlinear Algebraic Equations F(x) = 0, using ẋ = λ [αR + βP] or ẋ = λ [αF + βP *] R is a normal to a hyper-surface function of F, P normal to R, and P * normal to F

AU - Liu, Chein Shan

AU - Dai, Hong Hua

AU - Atluri, Satya N.

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2011

Y1 - 2011

N2 - To solve an ill-(or well-) conditioned system of Nonlinear Algebraic Equations (NAEs): F(x) = 0, we define a scalar hyper-surface h(x,t) = 0 in terms of x, and a monotonically increasing scalar function Q(t) where t is a time-like variable. We define a vector R which is related to ∂h/∂x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = αR + βP where α and β are optimized for fastest convergence. Using this ODV [x = λu], we derive an Optimal Iterative Algorithm (OIA) to solve F(x) = 0. We also propose an alternative Optimal Descent Vector [u = αF + βP *] where P * is normal to F. We demonstrate the superior performance of these two alternative OIAs with ODVs to solve NAEs arising out of the weak-solutions for several ODEs and PDEs. More importantly, we demonstrate the applicability of solving simply, most efficiently, and highly accurately, the Duffing equation, using 8 harmonics in the Harmonic Balance Method to find a weak-solution in time.

AB - To solve an ill-(or well-) conditioned system of Nonlinear Algebraic Equations (NAEs): F(x) = 0, we define a scalar hyper-surface h(x,t) = 0 in terms of x, and a monotonically increasing scalar function Q(t) where t is a time-like variable. We define a vector R which is related to ∂h/∂x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = αR + βP where α and β are optimized for fastest convergence. Using this ODV [x = λu], we derive an Optimal Iterative Algorithm (OIA) to solve F(x) = 0. We also propose an alternative Optimal Descent Vector [u = αF + βP *] where P * is normal to F. We demonstrate the superior performance of these two alternative OIAs with ODVs to solve NAEs arising out of the weak-solutions for several ODEs and PDEs. More importantly, we demonstrate the applicability of solving simply, most efficiently, and highly accurately, the Duffing equation, using 8 harmonics in the Harmonic Balance Method to find a weak-solution in time.

KW - Invariant-manifold

KW - Nonlinear Algebraic Equations

KW - Optimal Descent Vector (ODV)

KW - Optimal Iterative Algorithm (OIA)

KW - Optimal Iterative Algorithm with an Optimal Descent Vector (OIA/ODV)

KW - Perpendicular operator

UR - http://www.scopus.com/inward/record.url?scp=84863230414&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84863230414

VL - 81

SP - 335

EP - 362

JO - CMES - Computer Modeling in Engineering and Sciences

JF - CMES - Computer Modeling in Engineering and Sciences

SN - 1526-1492

IS - 3-4

ER -