LU factorization, Separable nonlinear equations
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.
ISRN Mathematical Analysis
Article ID 258072
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Published by Hindawi Publishing Corporation
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
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