Computing the Numerical Nullity of Sylvester Matrix of Univariate Polynomials

• Main Authors: Sung-chen Hsieh, 謝松臻
• Other Authors: Tsung-Lin Lee
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/28922594897287071233

碩士 === 國立中山大學 === 應用數學系研究所 === 103 === Computing the greatest common divisor (GCD) of univariate polynomials is one of the fundamental algebraic problems with a long history. The classical Euclidean algorithm is not suitable for practical numerical computation because computing GCD is an ill-posed problem in the sense that it is extremely sensitive to the data perturbations. In this thesis we study the QRGCD method, included in the SNAP package of Maple 9, which based on the theorem saying that the last nonzero row of R in the QR-factorization of Sylvester matrix provides a GCD of polynomials. The method works under the assumption that the lower right block of R is the zero matrix of the size of its nullity. However, the QR-factorization of a rank-deficient matrix is not unique, which implies that the lower right block may not be zero. Hence, we consider the rank-revealing QR-factorization algorithm (RRQR) to circumvent this situation.