Numerical linear algebra: algorithms, analysis and applications

数值线性代数：算法、分析与应用

• 批准号: 9236-2006
• 负责人: Paige, Christopher
• 金额: $ 2.77万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=338362
• 关键词: Numerical; linear; algebra; algorithms; analysis

In order to solve many scientific, engineering, business and other problems using the computer, we usually convert the problems to subproblems involving matrices --- rectangular arrays of numbers.  Because of this, matrix computations and its theory (often called numerical linear algebra) lies at the heart of most scientific computing.  The key to solving many smaller matrix problems is to find some appropriate factorizations of matrices --- so for example a given matrix can be described as the product of two or more such factors.  However many problems are large or complex, and the resulting matrix problems are extremely large --- a million rows and a million columns is not at all unusual.  And such large matrices are nearly always very sparse --- the vast majority of elements being zero.  Factorization methods can then become too costly, or sometimes simply impossible, and iterative methods may be necessary for solving such problems.  This research develops numerically reliable and efficient algorithms for solving both classes of problems.  It carries out analyses of such algorithms to prove their efficiency and reliability, and sensitivity analyses to show what effects changes in the data (for example caused by finite precision computation, or uncertainty in the data) will have on the final results. These analyses lead to greater understanding of individual problems and their computed answers, and improved general purpose and specific area algorithms. The results of the research are useful for all practitioners in scientific computation.  Some application areas of particular importance in Canada that use these techniques are the aerospace industry (via global positioning systems, aircraft body design, etc.), power generation and distribution, and resource discovery, extraction and transportation.

为了使用计算机解决许多科学、工程、商业和其他问题，我们通常将问题转化为涉及矩阵的子问题--数字的矩形阵列。矩阵计算及其理论（通常称为数值线性代数）是大多数科学计算的核心。解决许多较小矩阵问题的关键是找到矩阵的适当分解-- 例如，一个给定的矩阵可以被描述为两个或多个这样的因子的乘积。然而，许多问题是大的或复杂的，并且由此产生的矩阵问题是非常大的--一百万行和一百万列根本不是不寻常的。并且这样的大矩阵几乎总是非常稀疏的--绝大多数元素为零。因式分解方法可能变得过于昂贵，本文通过对这两类问题的数值分析，证明了算法的有效性和可靠性，并通过敏感性分析，说明了数据变化对算法性能的影响（例如由有限精度计算或数据中的不确定性引起的）将对最终结果产生影响。这些分析导致更好地理解个别问题及其计算答案，并改进通用和特定领域的算法。研究结果对科学计算领域的所有从业者都很有用。在加拿大，使用这些技术的一些特别重要的应用领域是航空航天工业（通过全球定位系统，飞机机身设计等），发电和配电以及资源发现、开采和运输。