Simpson's Double Integral Numerical Algorithm

Simpson's double integral numerical algorithm is a numerical integration method based on Simpson's rule, designed to compute definite integrals over two-dimensional regions. This algorithm achieves high accuracy and efficiency by partitioning the integration domain into small subregions and applying Simpson's formula to approximate the integral value on each subregion.

### Core Algorithm Concept Domain Discretization: The integration region is first divided into an even number of equally spaced subintervals along both x and y directions, forming a regular grid. Increasing the number of subintervals in each direction typically improves accuracy but also increases computational cost. Simpson's Rule Application: On each subinterval, Simpson's formula is applied for approximation. This formula uses quadratic polynomials to fit the integrand function, yielding higher precision than the trapezoidal rule on small intervals. Composite Simpson's Rule: The Simpson approximations from each direction are combined to form a composite Simpson formula for double integrals. The final result is obtained through a weighted summation of integral values from all subregions.

### Error Analysis The error in Simpson's double integral method primarily stems from higher-order derivatives of the integrand function. If the integrand is sufficiently smooth within the integration domain (e.g., with continuous fourth-order derivatives), the algorithm can achieve an error of order O(h^4), where h represents the step size of subintervals.

### Implementation Considerations This algorithm is particularly suitable for integrand functions that exhibit smooth variations across the integration domain. For functions with severe oscillations or singularities near certain points, adaptive integration methods or other numerical techniques may be required to enhance accuracy.

Simpson's double integral numerical algorithm finds extensive applications in scientific computing, engineering simulations, and financial modeling, serving as an effective tool for solving complex integration problems.