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Separable nonlinear equations have the form 𝐹(𝑦, 𝑧) ≑ 𝐴(𝑦)𝑧 + 𝑏(𝑦) = 0, where the matrix 𝐴(𝑦) ∈ Rπ‘šΓ—π‘ and the vector 𝑏(𝑦) ∈ Rπ‘š are continuously differentiable functions of 𝑦 ∈ R𝑛 and 𝑧 ∈ R𝑁. We assume that π‘š β‰₯ 𝑁 + 𝑛, and 𝐹'(𝑦, 𝑧) has full rank. We present a numerical method to compute the solution (π‘¦βˆ—, π‘§βˆ—) for fully determined systems (π‘š = 𝑁+ 𝑛) and compatible overdetermined systems (π‘š > 𝑁+ 𝑛). Our method reduces the original system to a smaller system 𝑓(𝑦) = 0 of π‘š βˆ’ 𝑁 β‰₯ 𝑛equations in 𝑦 alone. The iterative process to solve the smaller system only requires the LU factorization of one π‘šΓ— π‘š matrix per step, and the convergence is quadratic. Once π‘¦βˆ— has been obtained, π‘§βˆ— is computed by direct solution of a linear system. Details of the numerical implementation are provided and several examples are presented.

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ISRN Mathematical Analysis




Article ID 258072

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Copyright Β© 2013 Y.-Q. Shen and T. J. Ypma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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