Ill-conditioned Properties and hybrid computations

Approximate algebraic computation (AAC) has been one of the most important research areas in algebraic computation. The basis of AAC is an algorithm of computing approximate greatest common divisors (AppGCD) proposed by T. Sasaki and the author. AppGCD and its applications work well, especially, for obtaining accurate results of ill-conditioned problems. Algorithms and implementation methods of AppGCD are briefly surveyed and its applications such as hybrid integral, hybrid rational function approximation (HRFA), data smoothing by using HRFA and new hybrid method for computing Cauchy principal value integral are described. Further, a pathological feature of HRFA and relations of HRFA and ill-conditioned problems, and their applications are discussed.