Recursive Least Squares Estimation for Polynomial Coefficients in y(i) = b0 + b1 * x + b2 * x^2 + ... + bn * x^n

This text explores the application of recursive least squares (RLS) method for estimating coefficients b0, b1, ..., bn in the polynomial y(i) = b0 + b1 * x + b2 * x^2 + b3 * x^3 + ... + bn * x^n. The recursive least squares algorithm is a powerful numerical analysis technique particularly suitable for real-time parameter estimation in polynomial function fitting. The implementation typically involves initializing a covariance matrix and parameter vector, then recursively updating them using new data points through matrix operations. Key computational aspects include handling the regression vector φ = [1, x, x², ..., xⁿ] and updating the gain matrix K(k) = P(k-1)φ(k)/(λ + φ'(k)P(k-1)φ(k)), where λ represents the forgetting factor. Through iterative coefficient adjustment and polynomial characteristic analysis, RLS enables dynamic model refinement that enhances data interpretation and predictive capabilities. This method proves particularly valuable for adaptive systems and online parameter estimation, providing an efficient computational framework for exploring data patterns and deriving meaningful conclusions from polynomial relationships.