Abstract: Similar to having done for the mid-point and trapezoid quadrature rules, we obtain alternative estimations of error bounds for the Simpson’s quadrature rule involving n-time(1 ≤ n ≤ 4) differentiable mappings and then to the estimations of error bounds for the adaptive Simpson’s quadrature rule.

Key words: n-numerical quadrature rule, Simpson’s quadrature rule, adaptive quadrature
rule, differentiable mapping, error bounds