论文标题
无分矩阵分解的常见因素
Common Factors in Fraction-Free Matrix Decompositions
论文作者
论文摘要
我们使用精确的计算来考虑LU和QR矩阵分解。我们表明，无分数的高斯 - 巴雷斯的还原导致三角矩阵具有非平凡数量的公共行因子。我们确定了两种常见因素：系统和统计。系统因素取决于减少过程，而与数据无关，而统计因素取决于特定数据。我们将LU分解中的行因子的存在与史密斯 - jacobson矩阵的正常形式中出现的因素联系起来。对于统计因素，我们确定了创建它们的一些机制，并估算出其发生频率。类似的观察结果适用于无QR分解中的共同因素。我们的结论经过实验测试。
We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauss--Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith--Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.
