Curve Fitting Using Aitken Interpolation Method

Aitken interpolation is an efficient numerical analysis method commonly used for curve fitting and function approximation problems. This approach progressively refines polynomial interpolation results through an iterative algorithm to achieve higher precision approximations, particularly suitable for fitting discrete data points. The implementation typically involves recursive computation of interpolation coefficients with O(n²) time complexity.

During the Aitken interpolation process, initial interpolation points are systematically optimized through successive iterations, enabling the final polynomial function to better approximate real data. A key characteristic is its ability to reduce errors iteratively, thereby improving fitting accuracy. The algorithm can be implemented using nested loops to calculate divided differences, with convergence checks at each iteration stage. This method finds broad applications in engineering computations, data processing, and scientific simulations.

A typical application scenario is clock error fitting in satellite navigation systems, where precise clock error modeling is crucial. Aitken interpolation effectively fits clock error data by implementing progressive polynomial refinement, providing high-precision prediction results that ensure navigation system stability. Compared to other interpolation methods like Lagrange or Newton interpolation, Aitken's method demonstrates superior computational efficiency and convergence properties in practical implementations, often requiring fewer iterations to achieve target precision.

In summary, Aitken interpolation serves as a powerful and flexible numerical tool, particularly suitable for curve fitting problems demanding high accuracy. Code implementation typically involves initializing with base points, followed by iterative refinement steps that update interpolation coefficients until convergence criteria are met.