# AN EFFICIENT FOUR-POINT QUADRATURE SCHEME FOR RIEMANN-STIELTJES INTEGRAL

#### DOI NO:

https://doi.org/10.26782/jmcms.2021.01.00004

#### Keywords:

Quadrature rule,Riemann-Stieltjes,Simpson’s 3/8 rule,Composite form, Local error,Global error,Cost-effectiveness,Time-efficiency,

#### Abstract

In this work, a new four-point quadrature scheme is proposed for efficient approximation of the Riemann-Stieltjes integral (RS-integral). The composite form of the proposed scheme is also derived for the RS-integral from the concept of precision. Theoretically, the theorems related to the basic form, composite form, local and global errors of the new scheme are proved on the RS-integral. The correctness of the new proposed scheme is checked by g(t) = t, which reduces the proposed scheme into the original form of Simpson’s 3/8 rule for Riemann integral. The efficiency of the new proposed scheme is demonstrated by experimental work using programming in MATLAB against existing schemes. The order of accuracy and computational cost of the new proposed scheme is computed. The average CPU time is also measured in seconds. The obtained results demonstrate the efficiency of the proposed scheme over the existing schemes.

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