An adaptive hybrid quadrature scheme: Combining Simpson’s rule and Gaussian quadrature for enhanced numerical integration

Abadi Abraha Asgedom and Yohannes Yirga Kefela

PLOS ONE, 2026, vol. 21, issue 2, 1-18

Abstract: This study develops a novel adaptive hybrid quadrature that combines Simpson’s 1/3 rule with Gauss-Legendre quadrature to overcome the classical difficulties in performing numerical integration. Classical methods may encounter challenges in achieving a good balance between computational cost and precision, especially when it comes to functions characterized by strongly varying behaviors across their domains. We address these issues via an intelligent adaptation mechanism that reallocates computing resources dynamically on localized function features. We rigorously analyse its convergence properties analytically and prove optimal error estimates in the sense of fourth order accuracy with a strong performance improvement. The hybrid error estimation methodology is based on the mathematical inconsistency of polynomial interpolation and orthogonal polynomial approximation which provides an effective device for local error evaluation. Extensive numerical results indicate that the proposed scheme is consistently better than several existing schemes with significant reduction in function evaluations and acceptable accuracy for different test functions. The proposed framework reduces computational costs by up to 62% when compared to traditional adaptive methods. It maintains similar precision. We carefully examine implementation details, complexity analysis, and practical deployment factors. This work is particularly relevant for scientific computing applications that require high-precision integration in computational physics, engineering simulations, and financial mathematics.

Date: 2026
References: Add references at CitEc
Citations:

Downloads: (external link)
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0335582 (text/html)
https://journals.plos.org/plosone/article/file?id= ... 35582&type=printable (application/pdf)

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:plo:pone00:0335582

DOI: 10.1371/journal.pone.0335582

Access Statistics for this article

More articles in PLOS ONE from Public Library of Science
Bibliographic data for series maintained by plosone ().