Numerically-robust O(N2) RLS algorithms using least-squares prewhitening

We derive two new O(N²) algorithms for arbitrary recursive least-squares (RLS) estimation tasks. The algorithms employ a novel update for an inverse square-root factor of the exponentially-windowed input signal autocorrelation matrix that is the least-squares equivalent of a natural gradient prewhitening algorithm. Both of the new RLS algorithms require 4N²+O(N) multiply/adds, two divides, and one square root per iteration to implement. We can prove that our new algorithms are numerically-robust, and simulations are used to indicate this fact in fixed-point arithmetic. An algorithm that computes the square-root factorization of the input signal autocorrelation matrix is also described