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arXiv · 2603.16644

Perturbation Analysis for Preconditioned Normal Equations in Mixed Precision

James E. Garrison·Ilse C. F. Ipsen
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Abstract

For real matrices of full column-rank, we analyze the conditioning of several types of normal equations that are preconditioned by a randomized preconditioner computed in lower precision. These include symmetrically preconditioned normal equations, half-preconditioned normal equations, seminormal equations and not-normal equations. Our perturbation bounds are realistic and informative, and suggest that the conditioning depends only mildly on the quality of the preconditioner; however, it does depend on the size of the least squares residual -- even if the normal equations do not originate from a least squares problem. We illustrate that a randomized preconditioner can deliver a solution accuracy comparable to that of Matlab's mldivide command, is efficient in practice, and well-suited to GPU implementations. For the computation of the preconditioner, we propose an automatic selection of the precision, based on a fast condition number estimation in lower precision.

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James E. Garrison, Ilse C. F. Ipsen. 2026-03-17. Perturbation Analysis for Preconditioned Normal Equations in Mixed Precision. https://arxiv.org/abs/2603.16644

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