Perturbation Approach to Polynomial Root Estimation and Expected Maximum Modulus of Zeros with Uniform Perturbations

Ibrahim A. Nafisah, Sajad A. Sheikh, Mohammed A. Alshahrani, Mohammed M. A. Almazah, Badr Alnssyan and Javid Gani Dar ()
Additional contact information
Ibrahim A. Nafisah: Department of Statistics and Operations Research, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
Sajad A. Sheikh: Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, India
Mohammed A. Alshahrani: Department of Mathematics, College of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
Mohammed M. A. Almazah: Department of Mathematics, College of Sciences and Arts (Muhyil), King Khalid University, Muhyil 61421, Saudi Arabia
Badr Alnssyan: Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
Javid Gani Dar: Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed University), Pune 412115, India

Mathematics, 2024, vol. 12, issue 19, 1-16

Abstract: This paper presents a significant extension of perturbation theory techniques for estimating the roots of polynomials. Building upon foundational results and recent work by Pakdemirli and Yurtsever, as well as taking inspiration from the concept of probabilistic bounds introduced by Sheikh et al., we develop and prove several novel theorems that address a wide range of polynomial structures. These include polynomials with multiple large coefficients, coefficients of different orders, alternating coefficient orders, large linear and constant terms, and exponentially decreasing coefficients. Among the key contributions is a theorem that establishes an upper bound on the expected maximum modulus of the zeros of polynomials with uniformly distributed perturbations in their coefficients. The theorem considers the case where all but the leading coefficient receive a uniformly and independently distributed perturbation in the interval [ − 1 , 1 ] . Our approach provides a comprehensive framework for estimating the order of magnitude of polynomial roots based on the structure and magnitude of their coefficients without the need for explicit root-finding algorithms. The results offer valuable insights into the relationship between coefficient patterns and root behavior, extending the applicability of perturbation-based root estimation to a broader class of polynomials. This work has potential applications in various fields, including random polynomials, control systems design, signal processing, and numerical analysis, where quick and reliable estimation of polynomial roots is crucial. Our findings contribute to the theoretical understanding of polynomial properties and provide practical tools for engineers and scientists dealing with polynomial equations in diverse contexts.

Keywords: polynomial roots; perturbation theory; simulation; statistical model; random polynomial; numerical stability; uniform distribution; expectation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/19/2993/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/19/2993/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:19:p:2993-:d:1486030

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().