TITLE:
A Matrix-Based Reformulation of the Composite Simpson Rule for Double Numerical Integration over Rectangular Domains with Non-Uniformly Spaced Data
AUTHORS:
Lizandro B. R. Zegarra, Luis E. G. Armas, César C. Campos, Pedro G. R. Carrera, Herón J. M. Marchena, Carlos A. M. Alvarado, María E. B. Gemin, Aída B. D. Reyna
KEYWORDS:
Matrix Formulation of Numerical Integration, Double Numerical Integration, Composite Simpson’s 1/3 Rule
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
26,
2026
ABSTRACT: This work presents a matrix-based reformulation of the composite Simpson’s 1/3 rule for the numerical evaluation of double integrals over rectangular domains with continuous integrands and prescribed non-uniformly spaced nodes. The classical formulation is usually expressed through nested summations or through a matrix of order
(
n+1
)×(
m+1
)
, whose size increases with the number of discretization nodes. The proposed approach reorganizes the contributions of the subrectangles into a fixed 3 × 3 matrix whose nine entries are defined by weighted double summations over the computational mesh. Consequently, the numerical approximation can be written compactly as the product of matrices of orders 1 × 3, 3 × 3, and 3 × 1, with the vectors containing the Simpson coefficients
(
1,4,1
)
. The fixed-size representation provides a compact organization of the integration procedure and can reduce the auxiliary memory required for the matrix representation, while the asymptotic computational complexity remains
O(
nm
)
. Numerical results for non-uniform meshes show that the formulation provides accurate approximations consistent with the behavior of the composite Simpson’s 1/3 rule. The proposed method is intended as a simple and flexible alternative for double numerical integration when data are available on prescribed non-uniform rectangular meshes.