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Paper
Arrowhead Factorization of Real Symmetric Matrices and its Applications in Optimized Eigendecomposition
Presenter
Marcel Ferrari's first contact with computational science was in high school, where his interest in molecular dynamics simulations sparked his journey into the world of High-Performance Computing (HPC). This early exposure fuelled his pursuit of a Bachelor's in Computational Science and Engineering at ETH Zürich, with a focus in Geophysics. Collaborating with ETHZ's Theoretical Quantum Dynamics group, Marcel's Bachelor's thesis explored numerical linear algebra methods applied to instanton theory. Currently completing his Master's, Marcel is specialising in geophysics and robotics. His research interests include domain science, numerical methods and machine learning, with a particular fascination for the application of ML techniques in scientific fields. Alongside, he contributes as a Software Engineer at the Swiss National Supercomputing Center CSCS.
Description
This work introduces a new matrix decomposition, that we termed arrowhead factorization (AF). We showcase its applications as a novel method to compute all eigenvalues and eigenvectors of certain symmetric real matrices in the class of generalized arrowhead matrices. We present a clear definition and proof by construction of the existence of AF, detailing how to bridge the gap to full eigendecomposition. Our proposed method was tested against state-of-the-art routines, implemented in OpenBLAS, AOCL and Intel oneAPI MKL, using three synthetic benchmarks inspired by real world scientific applications. These experiments highlighted up to 49x faster runtimes, proving the validity and efficacy of our approach. Furthermore, we applied our method to a practical scenario by conducting a numerical experiment on simulation data derived from Golden-rule instanton theory. This real world application showed a performance gain ranging from 2.5×, for exact eigendecomposition, to over 38× with the most aggressive approximation strategy, underscoring the efficiency, robustness and flexibility of our algorithm.