对单数矩阵笔的一类随机数值方法的自值条件数的分析
Daniel Kressner1, Bor Plestenjak2
1Institute of Mathematics, EPFL, 1015 Lausanne, Switzerland.
概括
随机修改可以将单数矩阵笔转换为正则矩阵笔,为一般化自值问题提供稳定的数值解决方案. 复杂的随机矩阵增强了真实问题的数值稳定性.
科学领域:
- 数字分析 数字分析
- 线性代数 线性代数
- 计算数学 计算数学 计算数学
背景情况:
- 用单数矩阵笔解决一般化自身价值问题,由于不连续的自身价值,在数值上具有挑战性.
- 传统方法涉及在应用像QZ算法这样的解决方法之前,通过楼梯形式提取正规部分.
- 最近的研究探讨了随机修改,以将单一的笔转变为常规的笔.
研究的目的:
- 分析三种随机方法 (Hochstenbach,Mehl,Plestenjak) 用于将单数矩阵笔转换为正则的方法.
- 为了评估数字稳定性和转换笔的状态号码.
- 在这些转换中比较实数与复杂随机矩阵的有效性.
主要方法:
- 对三种随机修改技术的分析:修改,投影和增强.
- 使用正常等级来确保保留有限的固有值.
- 转换笔的固有值条件数与原始-弱条件数的比较.
- 实数对复杂随机矩阵对数值稳定性的影响的研究.
主要成果:
- 分析的随机方法将单数笔转换为正数笔,而不会改变有限的固有值.
- 转换的笔表现出与原来的-弱条件数字相比的自值条件数,表明有利的数值稳定性.
- 复杂的随机矩阵对于数值稳定性是可取的,即使使用真实笔和固有值.
- 对于泛化β和库马拉斯瓦米分布式随机变量的产物来说,获得了尖的左尾边界.
结论:
- 随机修改提供了一个数值稳定的替代方案,用于用单数矩阵笔解决通用自值问题.
- 弱自值条件数是检测简单有限自值的可靠指标.
- 使用复杂的随机矩阵可以提高这些方法对实值问题的数值稳定性.
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