AitkenDeltaSqu

AitkenDeltaSqu is a numerical method used to accelerate the convergence of iterative algorithms, particularly those used to find roots of equations or solve systems of equations. The method is named after Alexander Aitken, who developed it in the early 20th century. It is often applied to sequences generated by iterative methods such as the Newton-Raphson method or fixed-point iteration.

The AitkenDeltaSqu method works by transforming a slowly converging sequence into a new sequence that converges more rapidly. Given a sequence {x_n}, the AitkenDeltaSqu method constructs a new sequence {y_n} using the formula:

y_n = x_n - (x_{n+1} - x_n)^2 / (x_{n+2} - 2x_{n+1} + x_n)

This formula is derived by assuming that the sequence {x_n} can be approximated by a quadratic polynomial, and then using the polynomial to estimate the limit of the sequence.

The AitkenDeltaSqu method is particularly useful when the original sequence converges linearly, meaning that the error decreases by a constant factor at each step. In such cases, the AitkenDeltaSqu method can often transform the sequence into one that converges quadratically, where the error decreases by a factor that is itself decreasing.

However, the AitkenDeltaSqu method is not a panacea. It requires that the original sequence be sufficiently smooth and that the limit of the sequence be approached from a single side. If these conditions are not met, the method may not work as intended or may even diverge.

In summary, AitkenDeltaSqu is a powerful tool for accelerating the convergence of iterative algorithms, but it should be used with caution and understanding of its limitations.