A multi-step Ulm-Chebyshev-like method for solving nonlinear operator equations

Wei Ma; Ming Zhao; Jiaxin Li; Wei Ma; Ming Zhao; Jiaxin Li

doi:10.3934/math.20241389

AIMS Mathematics

2024, Volume 9, Issue 10: 28623-28642. doi: 10.3934/math.20241389

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Research article

A multi-step Ulm-Chebyshev-like method for solving nonlinear operator equations

Wei Ma ^{1
,
,},
Ming Zhao ²,
Jiaxin Li ¹

1.
School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China
2.
Public Basic Teaching Department, Henan Police College, Zhengzhou 450000, China

Received: 01 August 2024 Revised: 21 September 2024 Accepted: 26 September 2024 Published: 09 October 2024
MSC : 47H30, 65H10, 65J15

In this paper, based on the Ulm-Chebyshev iterative procedure, we present a multi-step Ulm-Chebyshev-like method to solve systems of nonlinear equations $ F(x) = 0 $,
$ \begin{equation*} \left\{\begin{array}{l} \quad {\bf{y}}_{n} = {\bf{x}}_{n}-B_{n}F( {\bf{x}}_{n}),\\ \quad {\bf z}_{n} = {\bf{y}}_{n}-B_{n}F( {\bf{y}}_{n}),\\ {\bf{x}}_{n+1} = {\bf z}_{n}-B_{n}F( {\bf z}_{n}),\\ \quad \bar{B}_{n} = 2B_{n}-B_{n}A_{n+1}B_{n},\\ B_{n+1} = \bar{B}_{n}+\bar{B}_{n}(2I-A_{n+1}\bar{B}_{n})(I-A_{n+1}\bar{B}_{n}),\quad n = 0,1,2,\ldots, \end{array}\right. \end{equation*} $
where $ A_{n+1} $ is an approximation of the derivative $ F'({\bf{x}}_{n+1}) $. This method does not contain inverse operators in its expression, and does not require computing Jacobian matrices for solving Jacobian equations. We have proved that the multi-step Ulm-Chebyshev-like method converges locally to the solution with $ R $-convergence rate 4 under appropriate conditions. Some applications are given, compared with other existing methods, where the most important features of the method are shown.
- nonlinear equation,
- multi-step,
- Ulm-Chebyshev-like method,
- $ R $-convergence rate 4
Citation: Wei Ma, Ming Zhao, Jiaxin Li. A multi-step Ulm-Chebyshev-like method for solving nonlinear operator equations[J]. AIMS Mathematics, 2024, 9(10): 28623-28642. doi: 10.3934/math.20241389

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Abstract

In this paper, based on the Ulm-Chebyshev iterative procedure, we present a multi-step Ulm-Chebyshev-like method to solve systems of nonlinear equations $ F(x) = 0 $,

$ \begin{equation*} \left\{\begin{array}{l} \quad {\bf{y}}_{n} = {\bf{x}}_{n}-B_{n}F( {\bf{x}}_{n}),\\ \quad {\bf z}_{n} = {\bf{y}}_{n}-B_{n}F( {\bf{y}}_{n}),\\ {\bf{x}}_{n+1} = {\bf z}_{n}-B_{n}F( {\bf z}_{n}),\\ \quad \bar{B}_{n} = 2B_{n}-B_{n}A_{n+1}B_{n},\\ B_{n+1} = \bar{B}_{n}+\bar{B}_{n}(2I-A_{n+1}\bar{B}_{n})(I-A_{n+1}\bar{B}_{n}),\quad n = 0,1,2,\ldots, \end{array}\right. \end{equation*} $

where $ A_{n+1} $ is an approximation of the derivative $ F'({\bf{x}}_{n+1}) $. This method does not contain inverse operators in its expression, and does not require computing Jacobian matrices for solving Jacobian equations. We have proved that the multi-step Ulm-Chebyshev-like method converges locally to the solution with $ R $-convergence rate 4 under appropriate conditions. Some applications are given, compared with other existing methods, where the most important features of the method are shown.

References

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