Romberg Integration Method

Romberg integration is an efficient numerical integration method that combines the advantages of Richardson extrapolation and the trapezoidal rule, achieving rapid convergence to high-precision integration results. For MATLAB beginners, mastering this algorithm provides insights into numerical integration principles while developing fundamental MATLAB programming skills.

### Core Algorithm Romberg method refines low-order approximations into high-order results by progressively subdividing the integration interval and applying extrapolation formulas. Key steps include: Initial trapezoidal sequence: Compute initial integration approximation using basic trapezoidal rule. Iterative refinement: Halve intervals recursively, leveraging previous results to minimize computations while enhancing accuracy through extrapolation formulas. Convergence checking: Terminate computation when error between successive results meets preset tolerance criteria.

### MATLAB Implementation Highlights Vectorization: Utilize MATLAB's matrix operations to replace inefficient loops for improved performance. Dynamic storage: Store intermediate results in tabular form (e.g., matrices) to facilitate extrapolation calculations. Adaptive termination: Implement relative or absolute error thresholds to balance precision and computational efficiency.

### Educational Value Implementing Romberg integration helps beginners deeply understand: Error reduction patterns in numerical integration How extrapolation techniques accelerate convergence MATLAB scripting and debugging methodologies

In practical applications, this method is widely used for engineering computations and scientific experimental data integration, serving as an essential introductory case for entering the field of numerical analysis.