Romberg extrapolation for Euler summation-based cubature on regular regions

Area of Science:

• Numerical Analysis
• Computational Mathematics

Background:

• Romberg extrapolation is a standard technique to accelerate the convergence of numerical integration methods, particularly the trapezoidal rule.
• Its direct application to multidimensional integration (cubature) is established for simple regions like hypercubes.

Purpose of the Study:

• To generalize Romberg extrapolation for Euler summation-based cubature.
• To apply this generalized method to arbitrary q-dimensional regular regions.
• To derive an explicit representation for the remainder term of this extended cubature method.

Main Methods:

• Formulation of Romberg extrapolation within the framework of Euler summation.
• Application to arbitrary q-dimensional regular regions.
• Derivation of the remainder term using mathematical analysis.

Main Results:

• Successful generalization of Romberg extrapolation for Euler summation-based cubature in q-dimensions.
• An explicit formula for the remainder term of the extended cubature method has been derived.
• The method is applicable to a broader class of regions beyond simple hypercubes.

Conclusions:

• The study provides a robust extension of Romberg extrapolation for multidimensional numerical integration.
• The derived remainder term offers valuable insights into the accuracy and error analysis of the proposed cubature method.
• This work advances the applicability of Romberg extrapolation in computational mathematics for complex regions.