AI 中文总结
本讲义是硕士水平高等数值线性代数课程的第一部分，介绍范数、稀疏矩阵、共轭梯度法等内容，配套Python代码复现示例，将算法应用于PDE、机器学习等领域。
AI 中文摘要
本讲义构成一门硕士水平高等数值线性代数课程的第一部分。其目的不仅是呈现经典算法，更要说明为何该学科比上一代重要得多。数值线性代数是随着偏微分方程（PDE）的数值求解发展起来的，长期以来，其大型稀疏系统都源自该领域。如今，对网络节点进行排序、将观测数据同化到天气预报中、对大型带噪数据集拟合模型，都会产生同类问题：规模过大无法分解、具有结构化特性，且只能通过矩阵-向量乘积访问。值得注意的是，所有这些问题仅需少数核心思路即可解决。因此，每章都会先展开一个标准主题，再将其应用到原领域之外。我们将讲解范数、矩阵分解、条件数与浮点算术；由有限差分、图和机器学习产生的稀疏矩阵；定常迭代与平滑性质；共轭梯度法和兰索斯法，以及谱聚类和早停正则化；阿诺尔迪法和GMRES法，以及PageRank和大型最小二乘问题；最后还会介绍预处理、施瓦茨区域分解和多重网格法。我们要求读者已修过线性代数入门课程。每节末尾会总结需掌握的内容，每章末尾附有习题，部分习题来自过往考试。配套的Python代码可复现数值示例。
英文摘要
These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why the subject has become considerably more central than it was a generation ago. Numerical linear algebra grew up alongside the numerical solution of partial differential equations, and for a long time that is where its large sparse systems came from. Ranking the nodes of a network, assimilating observations into a weather forecast, and fitting a model to a large noisy data set now lead to problems of the same kind: too large to factorise, structured, and accessible only through matrix-vector products. Strikingly few ideas are needed for all of them. Each chapter therefore develops a standard topic and then puts it to work outside its original setting. We treat norms, factorisations, conditioning and floating-point arithmetic; sparse matrices arising from finite differences, from graphs and from machine learning; stationary iterations and the smoothing property; the conjugate gradient and Lanczos methods, with spectral clustering and regularisation by early stopping; Arnoldi and GMRES, with PageRank and large least squares; and finally preconditioning, Schwarz domain decomposition and multigrid. We assume a first course in linear algebra. Every section closes with a summary of what should be retained and every chapter with exercises, several drawn from past examinations. Accompanying Python code reproduces the numerical illustrations.
Comments101 pages, 24 figures. Lecture notes; Part I of a two-part master's course. Generative AI (Claude Opus 5, Anthropic) was used to help identify seminal references, improve the language, and improve the graphical content; all statements, proofs, and references have been checked by the authors, who take full responsibility for the contents. Code: https://github.com/vicdolean/scicomp_examples
