Separable nonlinear equations have the form 𝐹(𝑦, 𝑧) ≡ 𝐴(𝑦)𝑧 + 𝑏(𝑦) = 0, where the matrix 𝐴(𝑦) ∈ R𝑚×𝑁 and the vector 𝑏(𝑦) ∈ R𝑚 are continuously differentiable functions of 𝑦 ∈ R𝑛 and 𝑧 ∈ R𝑁. We assume that 𝑚 ≥ 𝑁 + 𝑛, and 𝐹'(𝑦, 𝑧) has full rank. We present a numerical method to compute the solution (𝑦∗, 𝑧∗) for fully determined systems (𝑚 = 𝑁+ 𝑛) and compatible overdetermined systems (𝑚 > 𝑁+ 𝑛). Our method reduces the original system to a smaller system 𝑓(𝑦) = 0 of 𝑚 − 𝑁 ≥ 𝑛equations in 𝑦 alone. The iterative process to solve the smaller system only requires the LU factorization of one 𝑚× 𝑚 matrix per step, and the convergence is quadratic. Once 𝑦∗ has been obtained, 𝑧∗ is computed by direct solution of a linear system. Details of the numerical implementation are provided and several examples are presented.
Solving Separable Nonlinear Equations Using LU FactorizationISRN Mathematical Analysis
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Citation InformationShen, Yun-Qiu; Ypma, Tjalling J.: Solving Separable Nonlinear Equations Using LU Factorization. ISRN Mathematical Analysis Volume 2013 (2013), Article ID 258072, 5 pages.