Malaysian Journal of Mathematical Sciences, September 2026, Vol. 20, No. 3
Comparative Analysis of Trapezoidal and Simpson's Rule for Determining the Ellipsoid from its Polarization Tensor
Mohamad Yunos, N., Ahmad Khairuddin, T. K., Mohd Ali, N. M., and Ahmed Sukri, S.
Corresponding Email: taufiq@utm.my
Received date: 6 February 2025
Accepted date: 12 December 2025
Abstract:
The polarization tensor is essential for identifying and characterizing conducting objects, as it carries details about the object's size, material, and orientation. An ellipsoid may share the same first order polarization tensor as the actual object. Thus, investigating the first order polarization tensor for an ellipsoid can be beneficial in object characterization. Given the ellipsoidal first order polarization tensor, it is possible to determine the size of the ellipsoid, where the size of the ellipsoid is represented by its semi axes. Hence, this study proposes a comprehensive algorithm to determine an ellipsoid's semi axes from its first order polarization tensor. The methodology involves constructing a system of nonlinear equations using two numerical integration methods, the trapezoidal rule and Simpson's rule, followed by solving the system with Newton's method. Numerical examples are provided to validate the algorithm's reliability. The results indicate that, after reaching convergence, the error of the first order polarization tensor computed from the semi axes obtained using Simpson's rule is smaller than that of the trapezoidal rule. In addition, reducing the step size improves accuracy for both methods but increases computation time. Therefore, to achieve better accuracy while optimizing computation time, Simpson's rule is the preferred approach.
Keywords: numerical integrations; semi axes; matrices; depolarization factors; nonlinear equations
