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

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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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  1. Department of Mathematics, Western Washington University, Bellingham, WA, 98225-9063, USA

    Yunqiu Shen & Tjalling J. Ypma

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  1. Yunqiu Shen
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  2. Tjalling J. Ypma
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Correspondence to Tjalling J. Ypma.

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Shen, Y., Ypma, T.J. Numerical solution of separable nonlinear equations with a singular matrix at the solution. Numer Algor 85, 1195–1211 (2020). https://doi.org/10.1007/s11075-019-00861-0

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  • DOI: https://doi.org/10.1007/s11075-019-00861-0

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