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A numerical method to approximate the solutions of nonlinear systems in densifiable sets

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Abstract

In the present paper, we study the approximation of solutions (if any) of a system of equations, where the only condition required to the involved functions is to be continuous. To be more precise, by using the so called \(\alpha \)-dense curves, if the given system has some solution in a densifiable set we propose a method, which only use evaluations of single variable functions, to approximate at least one solution of the system in such set. The feasibility and reliability of the proposed method is illustrated by several examples, and its drawbacks are also discussed.

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Acknowledgements

To the anonymous reviewers for their helpful comments and suggestions which improved the quality of the paper. Also, to Prof. Dr. Gaspar Mora for his valuable help with the choice of the references cited in the paper.

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  1. Universidad Nacional de Educación a Distancia (UNED), Departamento de Matemáticas, CL. Candalix s/n, 03202, Elche, Alicante, Spain

    G. García

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Appendix

Appendix

Systems used in Sub-section 4.1

$$\begin{aligned} \text {Yamamoto1}\left\{ \begin{array}{ll} x_{1}^{2}+x_{2}^{3} =0\\ x_{2}^{2}=0 \end{array}\right. ,\quad \text {Powell1}\left\{ \begin{array}{ll} x_{1}-1 =0 \\ x_{1}x_{2}-1=0 \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Yamamoto2}\left\{ \begin{array}{ll} x_{1}^{3}+x_{1}x_{2}=0 \\ x_{2}+x_{2}^{2}=0 \end{array}\right. ,\quad \text {Fuchs}\left\{ \begin{array}{ll} x_{1}^{2}-x_{2}^{2}-1=0 \\ x_{2}^{2}+x_{2}^{2}-4=0 \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Brezinski1}\left\{ \begin{array}{ll} 0.5x_{2}^{2}-x_{2}^{2}=0 \\ -x_{2}+\sin (x_{1})+\sin (x_{2}-1)+1=0 \\ \end{array}\right. ,\quad \text {Bartish}\left\{ \begin{array}{ll} x_{1}^{2}+x_{2}^{2}-1=0 \\ 0.75 x_{1}^{3}-x_{2}=0 \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Powell2}\left\{ \begin{array}{ll} 10000x_{1}x_{2}-1=0 \\ \exp (-x_{1})+\exp (-x_{2})-1.0001=0 \\ \end{array}\right. ,\quad \text {Boggs}\left\{ \begin{array}{ll} x_{1}^{2}-x_{2}+1=0 \\ x_{1}-\cos (0.5\pi x_{2})=0 \\ \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Brezinski2}\left\{ \begin{array}{ll} -x_{1}+0.5x_{2}^{2}-1.5=0 \\ -x_{2}+0.605 \exp (1-x_{1}^{2})+0.395=0 \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Allgower-Geor1}\left\{ \begin{array}{ll} (x_{1}-x_{2}^{2})(x_{1}-\sin (x_{2}))=0 \\ \cos (x_{2})-x_{1})(\cos (x_{1})-x_{2})=0 \\ \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Yamamoto3}\left\{ \begin{array}{ll} x_{1}+x_{2}+x_{2}-1=0 \\ 0.2x_{1}^{3} +0.5x_{2}^{2}-x_{3}+0.5x_{3}^{3}+0.5= 0 \\ x_{2}+x_{2}+0.5x_{3}^{2}-0.5=0 \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Allgower-Geor2}\left\{ \begin{array}{ll} x_{1}^{2}+2x_{2}^{2}-4=0 \\ x_{1}^{2}+x_{2}^{2}+x_{3}-8=0 \\ (x_{1}-1)^{2}+(2x_{2}-\sqrt{2})^{2}+(x_{3}-5)^{2}-4=0 \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Brown-Conte}\left\{ \begin{array}{ll} 3x_{1}+x_{2}+2x_{3}^{2}-3=0 \\ -3x_{1}+5x_{2}^{2}+2x_{1}x_{3}-1=0 \\ 25x_{1}x_{2}+20x_{3}+12=0 \\ \end{array}\right. \end{aligned}$$
$$\begin{aligned} \text {Babitsch}\left\{ \begin{array}{ll} x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3}-47=0 \\ x_{1}+x_{2}^{2}-x_{3}^{2}=0 \\ (x_{3}-x_{1})(x_{3}-x_{2})-2=0 \end{array}\right. \end{aligned}$$

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García, G. A numerical method to approximate the solutions of nonlinear systems in densifiable sets. Numer Algor 96, 1925–1943 (2024). https://doi.org/10.1007/s11075-023-01690-y

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