Document Type
Article
Publication Date
2013
Keywords
LU factorization, Separable nonlinear equations
Abstract
Separable nonlinear equations have the form F(y,z) ≡ A (y)z + b(y) = 0, where the matrix A(y)∈ R m × N and the vector b(y) ∈ Rmare 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 ≥ n equations 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.
Publication Title
ISRN Mathematical Analysis
Volume
2013
Issue
Article ID 258072
DOI
http://dx.doi.org/10.1155/2013/258072
Required Publisher's Statement
Published by Hindawi Publishing Corporation
http://dx.doi.org/10.1155/2013/258072
Recommended Citation
Shen, Yun-Qiu; Ypma, Tjalling J.: Solving Separable Nonlinear Equations Using LU Factorization. ISRN Mathematical Analysis Volume 2013 (2013), Article ID 258072, 5 pages.
Subjects - Topical (LCSH)
Factorization (Mathematics); Differential equations, Nonlinear
Genre/Form
articles
Type
Text
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.
Language
English
Format
application/pdf
Comments
Copyright © 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.