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Scalable QR Factorisation of Ill-Conditioned Tall-and-Skinny Matrices on Distributed GPU Systems

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Publicado en:Mathematics vol. 13, no. 22 (2025), p. 3608-3629
Autor principal: Mijić Nenad
Otros Autores: Kaushik Abhiram, Živković Dario, Davidović Davor
Publicado:
MDPI AG
Materias:
Eigenvalues
Adaptive systems
Computation
Linear algebra
Matrices (mathematics)
Graphics processing units
Communication
Decomposition
Linear systems
Algorithms
Matrix algebra
Numerical stability
Stability
Critical path
Distributed memory
Orthogonality
Factorization
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Resumen:The QR factorisation is a cornerstone of numerical linear algebra, essential for solving overdetermined linear systems, eigenvalue problems, and various scientific computing tasks. However, computing it for ill-conditioned tall-and-skinny (TS) matrices on large-scale distributed-memory systems, particularly those with multiple GPUs, presents significant challenges in balancing numerical stability, high performance, and efficient communication. Traditional Householder-based QR methods provide numerical stability but perform poorly on TS matrices due to their reliance on memory-bound kernels. This paper introduces a novel algorithm for computing the QR factorisation of ill-conditioned TS matrices based on CholeskyQR methods. Although CholeskyQR is fast, it typically fails due to severe loss of orthogonality for ill-conditioned inputs. To solve this, our new algorithm, mCQRGSI+, combines the speed of CholeskyQR with stabilising techniques from the Gram–Schmidt process. It is specifically optimised for distributed multi-GPU systems, using adaptive strategies to balance computation and communication. Our analysis shows the method achieves accuracy comparable to Householder QR, even for extremely ill-conditioned matrices (condition numbers up to <inline-formula>1016</inline-formula>). Scaling experiments demonstrate speedups of up to <inline-formula>12×</inline-formula> over ScaLAPACK and <inline-formula>16×</inline-formula> over SLATE’s CholeskyQR2. This work delivers a method that is both robust and highly parallel, advancing the state-of-the-art for this challenging class of problems.
ISSN:2227-7390
DOI:10.3390/math13223608
Fuente:Engineering Database