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Solving Separable Nonlinear Equations Using LU Factorization
ISRN Mathematical Analysis
  • Yun-Qiu Shen, Western Washington University
  • Tjalling Ypma, Western Washington University
Document Type
Article
Publication Date
1-1-2013
Disciplines
Abstract

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.

Required Publisher's Statement

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.

Comments

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.

Creative Commons License
Creative Commons Attribution 3.0
Citation Information
Shen, Yun-Qiu; Ypma, Tjalling J.: Solving Separable Nonlinear Equations Using LU Factorization. ISRN Mathematical Analysis Volume 2013 (2013), Article ID 258072, 5 pages.