The Heronian and Harmonic Means Form the Basis of the Modified Trapezoid Method

Mayser Elan Abbas  AL-Owaidi; Maryam Alaa  Abdulhussein

Authors

Mayser Elan Abbas AL-Owaidi AL-Furat Al-Awsat Technical University (ATU), Technical Institute of Diwaniyah, Iraq
Maryam Alaa Abdulhussein AL-Furat Al-Awsat Technical University (ATU), Technical Institute of Diwaniyah, Iraq

Keywords:

Heronian imply, trapezoidal rule, harmonic imply

Abstract

Despite estimating an integral using a proper numerical integration technique seems trivial, classical numerical integration methods such as the trapezoidal rule sometimes yield larger approximation relative error and therefore, this study is motivated to provide a more accurate numerical integration techniques that are more suitable for scientific and engineering applications. The gap in knowledge is that existing mean-based quadrature methods are not accurate when a function behaves nonlinearly within one of the quadrature subintervals. In order to bridge the above gap, the study proposes a trapezoidal method assimilating both Heronian and harmonic means in the process of the evaluation of intermediate values. By several numerical examples, it is shown that the absolute errors of the proposed method are always less than the corresponding absolute errors generated by the classical trapezoidal rule and the method based on the arithmetic mean. These results show an improved representation of how functions vary and as such a better accuracy in integrals when Heronian and harmonic means are used together. This means that this approach could be a more robust numerical toolbox when it comes to applied mathematics, engineering, and scientific computing problems that need numerically accurate integral estimates

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