Householder Tridiagonalization as Preprocessing for the Jacobi Eigenvalue Algorithm
Year: 2026
Type: preprint
Abstract: Householder tridiagonalization is normally not used as preprocessing for the Jacobi eigenvalue algorithm because it costs O(n³) and subsequent Jacobi rotations gradually destroy the tridiagonal zeros. We show that this does not determine whether preprocessing is beneficial. Algebraically, Householder tridiagonalization concentrates all off-diagonal energy into the first off-diagonal band while also redistributing energy between the diagonal and off-diagonal parts of the matrix. This leads to favorable and unfavorable regimes related to diagonal spread. Experiments with NVIDIA cuSOLVER on six families of dense symmetric matrices show a consistent separation between five ordinary test families and one deliberately adverse unequal-variance covariance family. The eigenvalue-only experiment was performed on an RTX 3060, RTX 5060 Ti, and A100. Across all three GPUs, the complete Householder preprocessing + Jacobi pipeline was faster than standalone Jacobi for all five ordinary families from matrix size n = 128 onward. Averaged across these five families, the complete-pipeline runtime reduction ranged from 7.1% to 10.0% at n = 128, and from 20.8% to 31.6% at n = 1024. Repeating the RTX 3060 experiment with five different random seeds produced the same qualitative family-level behavior. A separate experiment requested eigenvectors in both paths and showed the same qualitative separation, although the additional eigenvector work shifted the crossover to larger matrix sizes. The unequal-variance covariance family remained unfavorable on all three GPUs. Its large initial diagonal spread was associated with substantial transfer of energy away from the diagonal, increased Jacobi work, and longer total runtime. The results suggest initial diagonal spread as an inexpensive warning that Householder preprocessing may be unfavorable.
Authors: Byazrova, Eteri, Morozov, Yuri
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204160596
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