Iterative approach for simultaneously finding roots of nonlinear equations with multiplicities

Monika Panwar; Sonia Bhalla; Higinio Ramos; Ramandeep Behl; Monika Panwar; Sonia Bhalla; Higinio Ramos; Ramandeep Behl

doi:10.3934/math.20261253

AIMS Mathematics

2026, Volume 11, Issue 9: 31833-31860. doi: 10.3934/math.20261253

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Iterative approach for simultaneously finding roots of nonlinear equations with multiplicities

1.
Department of Mathematics, Chandigarh University, Gharuan, Mohali, Punjab 140413, India
2.
Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain
3.
Escuela Polit'ecnica Superior, Universidad de Salamanca, Campus Viriato, 49029 Zamora, Spain
4.
Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received: 18 June 2026 Revised: 09 September 2026 Accepted: 11 September 2026 Published: 24 September 2026
MSC : 65B99, 65H05

This paper presents a two-step iterative approach for simultaneously approximating a finite number of roots of a nonlinear equation, where the multiplicities are assumed to be known. A convergence analysis of the proposed scheme is also presented. The proposed method achieves an order of convergence $ 4p $ for polynomial equations and $ 2p $ for non-polynomial equations when combined with any iterative method of order $ p $ for finding multiple roots. Computational results and error analysis graphs demonstrate that the special cases of the proposed scheme exhibit superior performance in terms of CPU execution time, residual error, and convergence rate.
- nonlinear equations,
- iterative methods,
- simultaneous multiple roots,
- convergence analysis
Citation: Monika Panwar, Sonia Bhalla, Higinio Ramos, Ramandeep Behl. Iterative approach for simultaneously finding roots of nonlinear equations with multiplicities[J]. AIMS Mathematics, 2026, 11(9): 31833-31860. doi: 10.3934/math.20261253

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Abstract

This paper presents a two-step iterative approach for simultaneously approximating a finite number of roots of a nonlinear equation, where the multiplicities are assumed to be known. A convergence analysis of the proposed scheme is also presented. The proposed method achieves an order of convergence $ 4p $ for polynomial equations and $ 2p $ for non-polynomial equations when combined with any iterative method of order $ p $ for finding multiple roots. Computational results and error analysis graphs demonstrate that the special cases of the proposed scheme exhibit superior performance in terms of CPU execution time, residual error, and convergence rate.

References

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