About a fixed-point-type transformation to solve quadratic matrix equations using the Krasnoselskij method

IF 1.8 3区数学 Q1 MATHEMATICS, APPLIED Mathematical Methods in the Applied Sciences Pub Date : 2022-05-03 DOI:10.1002/mma.8336

Miguel Ángel Hernández-Verón, Natalia Romero-Álvarez

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Abstract

In this paper, we study the simplest quadratic matrix equation: $Q (X) = X^{2} + B X + C = 0$ . We transform this equation into an equivalent fixed-point equation, and based on it, we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than the successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor–corrector iterative scheme, with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improve the numerical application of the Newton method when approximating a solution of the quadratic matrix equation.

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关于用Krasnoselskij方法求解二次矩阵方程的不动点型变换

本文研究了最简单的二次矩阵方程:Q (x) = x2 + bx + c = 0$$ \mathcal{Q}(X)={X}^2+ BX+C=0 $$。将该方程转化为等价的不动点方程，并在此基础上构造Krasnoselskij方法。通过这种变换，我们可以得到比逐次逼近法更精确的迭代格式。此外，在适当的条件下，我们用Krasnoselskij方法建立了该方程解的存在性和局域性的不同结果。最后，我们在数值上看到，以Krasnoselskij方法作为预测器，牛顿方法作为校正器的预测-校正迭代方案可以改善牛顿方法在近似二次矩阵方程解时的数值应用。

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.

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