Relaxed NewtonSLRA for Approximate GCD

Abstract

We propose a better algorithm for approximate GCD in terms of robustness and distance, based on the NewtonSLRA algorithm that is a solver for the structured low rank approximation (SLRA) problem. Our algorithm mainly enlarges the tangent space in the Newton-SLRA algorithm and adapts it to a certain weighted Frobenius norm. By this improvement, we prevent a convergence to a local optimum that is possibly far from the global optimum. We also propose some modification using a sparsity on the NewtonSLRA algorithm for the subresultant matrix in terms of computing time.

The GPGCD Algorithm with the Bézout Matrix for Multiple Univariate Polynomials

“…In both categories, algorithms based on various methods have been proposed including the Euclidean algorithm ( [1], [19], [20]), low-rank approximation of the Sylvester matrix or subresultant matrices ( [5], [6], [10], [11], [12], [21], [24], [26], [27], [29]), Padé approximation ( [17]), and optimizations ( [3], [13], [15], [22], [28]). Among them, the second author of the present paper has proposed the GPGCD algorithm based on low-rank approximation of subresultant matrices by optimization ( [24], [25], [26]), which belongs to the second category above.…”