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Numerical Algorithms
Published in: Numerical Algorithms 4/2020

04-01-2020 | Original Paper

Numerical solution of separable nonlinear equations with a singular matrix at the solution

Authors: Yunqiu Shen, Tjalling J. Ypma

Published in: Numerical Algorithms | Issue 4/2020

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Abstract

We present a numerical method for solving the separable nonlinear equation A(y)z + b(y) = 0, where A(y) is an m × N matrix and b(y) is a vector, with y ∈Rn and z ∈RN. We assume that the equation has an exact solution (y∗, z∗). We permit the matrix A(y) to be singular at the solution y∗ and also possibly in a neighborhood of y∗, while the rank of the matrix A(y) near y∗ may differ from the rank of A(y∗) itself. We previously developed a method for this problem for the case m = n + N, that is, when the number of equations equals the number of variables. That method, based on bordering the matrix A(y) and finding a solution of the corresponding extended system of equations, could produce a solution of the extended system that does not correspond to a solution of the original problem. Here, we develop a new quadratically convergent method that applies to the more general case m ≥ n + N and produces all of the solutions of the original system without introducing any extraneous solutions.
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Metadata
Title
Numerical solution of separable nonlinear equations with a singular matrix at the solution
Authors
Yunqiu Shen
Tjalling J. Ypma
Publication date
04-01-2020
Publisher
Springer US
Published in
Numerical Algorithms / Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00861-0

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