計算單變數多項式之最大公因式為一歷史悠久的基礎代數問題。計算最大公因式為一對
資料擾動極敏感的ill-posed problem ，這使得典型歐幾里得演算法並不適合作實際的
數值計算。在本文中我們探討Maple 9 中SNAP 套件QRGCD ，其理論立足於由西爾
維斯特矩陣所得正交化分解之上三角矩陣的最後一個非零列為最大公因式。此方法是作
用在一假設下：上三角矩陣的右下方陣為一零方陣其大小等同西爾維斯特矩陣矩陣之零
核維度。然而非滿秩矩陣的正交化分解並不唯一，這意味右下方陣可能不為零矩陣。最
後我們藉由rank-revealing QR-factorization 演算法得到最大公因式之次數。

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.

論文審定書i
摘要ii
Abstract iii
1 Introduction 1
2 Preliminaries 2
2.1 Sylvester matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Uniqueness of QR-factorization . . . . . . . . . . . . . . . . . . . . . 3
2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Algorithms for greatest common divisor 6
3.1 QR-factorization to compute the GCD . . . . . . . . . . . . . . . . . 6
3.2 Counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Rank revealing QR-factorization 12
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.1 Rank one deficiency . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.2 Rank deficiency more than one . . . . . . . . . . . . . . . . . 14
5 Numerical results 16
5.1 Experiments of RRQR . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Experiments of QRGCD . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2.1 Experiment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2.2 Experiment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2.3 Experiment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2.4 Experiment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Conclusion 20
Appendices 21
References 22
List of Tables
1 Singular matrices with and without using RRQR . . . . . . . . . . . 16
2 Singular matrices with and without using RRQR . . . . . . . . . . . 17
3 Finding GCD by QRGCD . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Finding GCD by QRGCD . . . . . . . . . . . . . . . . . . . . . . . . 18
List of Figures
1 The lower right block of R obtained by QR . . . . . . . . . . . . . . . 19
2 The lower right block of R obtained by RRQR . . . . . . . . . . . . . 19
3 The matrix of RΠ⊤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 The matrix of GRΠ⊤ . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 The lower right block of GRΠ⊤ . . . . . . . . . . . . . . . . . . . . . 20

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