A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

Salima Kouser; Shafiq Ur Rehman; Mabkhoot Alsaiari; Fayyaz Ahmad; Mohammed Jalalah; Farid A. Harraz; Muhammad Akram; Salima Kouser; Shafiq Ur Rehman; Mabkhoot Alsaiari; Fayyaz Ahmad; Mohammed Jalalah; Farid A. Harraz; Muhammad Akram

doi:10.3934/math.2024762

AIMS Mathematics

2024, Volume 9, Issue 6: 15782-15795. doi: 10.3934/math.2024762

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Research article

A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

1.
Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan
2.
Department of Chemistry, Faculty of Science and Arts at Sharurah, Najran University, Sharurah 68342, Saudi Arabia
3.
Promising Centre for Sensors and Electronic Devices (PCSED), Advanced Materials and Nano-Research Centre (AMNRC), Najran University, Najran 11001, Saudi Arabia
4.
Department of Applied Sciences, National Textile University, Faisalabad, Pakistan
5.
Department of Electrical Engineering, College of Engineering, Najran University, Najran 11001, Saudi Arabia
6.
Department of Engineering, University of Sannio, 82100, Benevento, Italy

Received: 31 January 2024 Revised: 12 April 2024 Accepted: 15 April 2024 Published: 30 April 2024
MSC : 65F15, 34L16, 34L20, 35P20, 35P15, 47A75

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $ \Omega $ and $ \Phi $, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $ T_{n}^{-1}(\Phi) T_n(\Omega) $, given a monotonic ratio $ r = \Omega/\Phi $. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.
- smoothing spline algorithm,
- preconditioned banded symmetric Toeplitz matrices,
- eigenvalues,
- polynomial interpolation
Citation: Salima Kouser, Shafiq Ur Rehman, Mabkhoot Alsaiari, Fayyaz Ahmad, Mohammed Jalalah, Farid A. Harraz, Muhammad Akram. A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices[J]. AIMS Mathematics, 2024, 9(6): 15782-15795. doi: 10.3934/math.2024762

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Abstract

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $ \Omega $ and $ \Phi $, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $ T_{n}^{-1}(\Phi) T_n(\Omega) $, given a monotonic ratio $ r = \Omega/\Phi $. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.

References

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