Verified Numerical Multiple Integration Based on Composite Gauss-Legendre Quadrature and Product Cubature Rules

Abstract

Abstract. We propose a verified numerical multiple integration based on the composite Gauss-Legendre quadrature whose division number of individual remainders is fewer than that of individual main terms. By combining remainder terms partially, the number of calculations for higher-order differentials, which take a long time to compute, can be reduced. Thus, a computation time of the proposed method decrease. In addition, to utilize a product cubature rule, the number of high-order differentials in remainder terms of multiple integration can be reduced. We illustrate the effectiveness of the proposed method using examples in two and three dimensions.