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.
Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential operators, where non-normality can lead to significant transient amplification and sensitivity to perturbations that are not captured by eigenvalue analysis alone. At large scales, commonly used numerical approaches for pseudospectra computation can become computationally demanding, as they require repeated auxiliary computations to identify spectrally sensitive regions in the complex plane. We present a neural network-based approach that predicts sensitive regions directly from matrix features, thereby avoiding exhaustive pseudospectra evaluation across the entire complex plane. We calibrate the prediction threshold on validation data to ensure reliable coverage of sensitive regions. The trained neural network guides the selection of grid points requiring full computation, enabling focused computation only where necessary. The approach provides a practical preprocessing strategy for efficient pseudospectra computation. Numerical experiments on non-normal banded matrices demonstrate substantial speedup compared to full grid-based numerical evaluation while maintaining high accuracy in identifying sensitive regions.
EigenDecomposition (ED) is at the heart of many computer vision algorithms and applications. One crucial bottleneck limiting its usage is the expensive computation cost, particularly for a mini-batch of matrices in deep neural networks. Our previous work proposed a dedicated QR-based ED algorithm for batched small matrices (dim${<}32$). This short paper targets the limitation and proposes a batch-efficient Divide-and-Conquer based ED algorithm for larger matrices. The numerical test shows that for a mini-batch of matrices whose dimensions are smaller than $64$, our method can be much faster than the Pytorch SVD function.