Numerical Integration Techniques: A Comprehensive Review
DOI:
https://doi.org/10.29070/cq10cp47Keywords:
Numerical integration, Gaussian quadrature, Monte Carlo methods, error analysis, adaptive algorithmsAbstract
When analytical solutions are not accessible, numerical integration allows one to approximate definite integrals, making it a key tool in mathematics and the applied sciences. From ancient methods like Newton-Cotes formulae to advanced approaches like Gaussian quadrature and current probabilistic methods like Monte Carlo integration, this study covers it all when it comes to major numerical integration techniques. Engineering, economics, and data science are just a few of the many areas that have found use for these methods, and the article delves into their theoretical underpinnings, error analysis, and convergence characteristics. Problems including computing efficiency, singularities, and high-dimensional integral processing are examined in detail. Future research prospects are further illuminated by discussing recent breakthroughs in adaptive algorithms and hybrid techniques. By compiling existing information and highlighting new developments in numerical integration, this study hopes to be a helpful resource for academics and industry professionals.
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