A new idea on the interval-symbol method with correct zero rewriting for reducing exact computations

Abstract

The ISCZ method (Interval-Symbol method with Correct Zero rewriting) was proposed in [2] based on Shirayanagi-Sweedler stabilization theory ([3]), to reduce the amount of exact computations as much as possible to obtain the exact results. The authors of [2] applied this method to Buchberger's algorithm which computes a Gröbner basis, but the effectiveness was not achieved except for a few examples. This is not only because the complex structure of Buchberger's algorithm causes symbols to significantly grow but also because we naively implemented the ISCZ method without any particular devices. In this poster, we propose a new idea for efficiency of the ISCZ method and show its effect by applying it to calculation of Frobenius canonical form of square matrices. Jordan canonical form is also well-known, but it requires an extension of the field containing the roots of its characteristic polynomial. On the other hand, Frobenius canonical form can be computed by using only basic arithmetic operations, but nevertheless has almost the same information as Jordan canonical form.