Minimum converging precision of the QR-factorization algorithm for real polynomial GCD

Authors:
Pramook Khungurn;Hiroshi Sekigawa;Kiyoshi Shirayanagi
Affiliations:
Massachusetts Institute of Technology, Cambridge, MA;Nippon Telegraph and Telephone Corporation, Kanagawa, Japan;Tokai University, Kanagawa, Japan
Venue:
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Year:
2007

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Abstract

Shirayanagi and Sweedler proved that a large class of algorithms over the reals can be modified slightly so that they also work correctly on fixed-precision oating-point numbers. Their main theorem states that, for each input, there exists a precision, called the minimum converging precision (MCP), at and beyond which the modified "stabilized" algorithm follows the same sequence of instructions as that of the original "exact" algorithm. Bounding the MCP of any non-trivial and useful algorithm has remained an open problem. This paper studies the MCP of an algorithm for finding the GCD of two univariate polynomials based on the QR-factorization. We show that the MCP is generally incom-putable. Additionally, we derive a bound on the minimal precision at and beyond which the stabilized algorithm gives a polynomial with the same degree as that of the exact GCD, and another bound on the minimal precision at and beyond which the algorithm gives a polynomial with the same support as that of the exact GCD.