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Arrowhead Factorization of Real Symmetric Matrices and its Applications in Optimized Eigendecomposition
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2024-01-01
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Conference Paper
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Abstract
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 2.5x, for exact eigendecomposition, to over 38x with the most aggressive approximation strategy, underscoring the efficiency, robustness and flexibility of our algorithm.
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PASC 2024: PROCEEDINGS OF THE PLATFORM FOR ADVANCED SCIENTIFIC COMPUTING CONFERENCE
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2024 Platform for Advanced Scientific Computing Conference-PASC
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Eigenvalue decomposition; Real symmetric matrix; Arrowhead matrix; Block Arrowhead matrix; Performance analysis