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# Computer Geometries for Finding All Real Zeros of Polynomial Equations Simultaneously

Naila Rafiq1, Saima Akram2, Mudassir Shams3,*, Nazir Ahmad Mir1

1 Department of Mathematics, NUML, Islamabad, 44000, Pakistan
2 Centre for Advanced Studies in Pure & Applied Mathematics, Bahauddin Zakariya University, Multan
3 Department of Mathematics and Statistics, Riphah International University I-14, Islamabad, 44000, Pakistan

* Corresponding Author: Mudassir Shams. Email:

(This article belongs to this Special Issue: Role of Computer in Modelling & Solving Real-World Problems)

Computers, Materials & Continua 2021, 69(2), 2635-2651. https://doi.org/10.32604/cmc.2021.018955

## Abstract

In this research article, we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation. We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab. Lower bound of convergence of simultaneous schemes is calculated using Mathematica. Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented. Some real life engineering applications along with some higher degree polynomials are considered as numerical test problems to show performance and efficiency of the derivative free family of numerical methods with comparison of an existing method of same order in literature. Local computational order of convergence, CPU time, graph of computational order of convergence and residual error graphs elaborate efficiency, robustness and authentication of the suggested family of numerical methods in its domain.

## Keywords

N. Rafiq, S. Akram, M. Shams and N. Ahmad Mir, "Computer geometries for finding all real zeros of polynomial equations simultaneously," Computers, Materials & Continua, vol. 69, no.2, pp. 2635–2651, 2021.

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