Iterative Solution of a System of Nonlinear Algebraic Equations F(x)=0, Using (x) over dot = lambda[alpha R plus beta P] or (x) over dot = lambda[alpha F plus beta P*] R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F

作者：Liu Chein Shan^*; Dai Hong Hua; Atluri Satya N

来源：Computer Modeling in Engineering & Sciences, 2011, 81(3-4): 335-362.

• Nonlinear algebraic equations
• Perpendicular operator
• Optimal Descent Vector (ODV)
• Optimal Iterative Algorithm (OIA)
• Optimal Iterative Algorithm with an Optimal Descent Vector (OIA/ODV)
• Invariant-manifold

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 partial derivative h/partial derivative x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = alpha R + beta P where alpha and beta are optimized for fastest convergence. Using this ODV [(x) over dot = lambda u], we derive an Optimal Iterative Algorithm (OIA) to solve F(x) = 0. We also propose an alternative Optimal Descent Vector [u = alpha F + beta P*] 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.

• 出版日期2011-11