A matrix QR-factorization approach to common factor extraction in the noisy data case

The obvious approach to computing the common factor (i.e., greatest common divisor (GCD)) between polynomials over a real number field is to employ Euclid´s algorithm. However, this algorithm is not robust if the polynomial coefficients are perturbed by noise. Here we see that GCD computation is equivalent to QR-factorizing a rank deficient near-to-Toeplitz matrix derived from the Sylvester matrix of the polynomials. Given noisy data the matrix is only nearly rank deficient. We summarize a computationally efficient and numerically reliable algorithm for QR-factorizing the nearly rank deficient matrix