Symbolic and Algebraic Computations is a multidisciplinary subject including many areas related to mathematically formulated computational problems (e.g. solving polynomial equations). In this workshop, we will have a discussion on Symbolic-Numeric Computations (with considerations on a priori or posteriori error) and Related Topics (Control Theory, Numerical Computations and so on).
Especially, we would like to discuss approximate GCD. Its history and future works are our interests. Any researcher interested in approximate GCD is welcome to join us though you may need some fundamentals on Symbolic and Algebraic Computations.
The followings may be useful for studying the fundamentals in Symbolic and Algebraic Computations.
In this talk, we introduce some fundamental properties of GCD (Euclid's algorithm, Half GCD, subresultant mapping and so on) in a (univariate or multivariate) polynomial ring over a field, the reason why the approximate GCD was studied and some major algorithms for approximate GCD (QRGCD, UVGCD, FastGCD, STLN based method and so on).
"Generalized KYP Lemma", proposed by Hara and Iwasaki, is a generalization of well-known "KYP Lemma" in control theory. In this presentation, we discuss an extension of the "Generalized KYP Lemma" to a parametric system.
When we solve the problem of computational geometry, it plays an important role to determine whether a point lies on, to the left of, or to the right of the oriented line defined by other points. This kind of judging is called The 2D Orientation Problem. The answer of this can be predicated by the sign of the determinant and we need the speed and the correctness of computation. In this talk, we introduce some known methods for this problem.
In this talk, we give an improved method for the orientation problem based on the known method.
For computing the approximate GCD in ill-conditioned cases (GCD with small leading-coefficients, with big leading-coefficients, and so on), their numerical computation will be unstable by the reason of instability even if it is known. In this talk, we introduce such ill-condition cases and propose some use of instability to compute approximate GCD, inversely.
A problem for calculating approximate GCD has attracted attentions as an optimization problem. In this talk, we focus on approximate GCD algorithm with optimization strategy and take a look at a glance at its development including the author's results.
Computing polynomial GCDs is one of important fundamental computations in algebraic computations hence there are many studies from the viewpoints of exact and (even) approximate computations. In this talk, we would like to have a discussion on the stability of approximate GCD in the near future, as an exact computation with some perturbations.