Under construction. This chapter is still being developed and may change as the analysis and experiments are refined.
The usual objection is that Householder tridiagonalization costs O(n³) and Jacobi rotations immediately create fill-in. This chapter asks a different question: whether the transformed starting matrix can nevertheless reduce later Jacobi work. It analyzes energy redistribution and parallel pivot structure, then tests six symmetric matrix families with NVIDIA cuSOLVER.
Tridiagonalization does not give Jacobi a permanently sparse matrix, but it changes where the matrix energy is located before the Jacobi iterations begin.
The chapter therefore reaches a conditional conclusion: Householder preprocessing can substantially help a parallel Jacobi eigensolver, but not for every matrix.
Because the possible benefit is not the permanent preservation of tridiagonal zeros. Tridiagonalization changes the starting distribution of diagonal and off-diagonal energy and concentrates all off-diagonal energy in the first band. That can give the standard first parallel Jacobi set a much stronger collection of pivots, although Householder similarity can also move energy away from the diagonal, so the effect is matrix-dependent.
Experiments with NVIDIA cuSOLVER on six families of dense symmetric matrices showed a consistent separation between five ordinary test families and one deliberately adverse unequal-variance covariance family.
In the eigenvalue-only experiments, Householder preprocessing reduced the subsequent Jacobi work for the five ordinary families. The experiment was run on an RTX 3060, RTX 5060 Ti, and A100; across all three GPUs, the complete preprocessing pipeline was faster than standalone Jacobi from matrix size n = 128 onward. Averaged across the five ordinary 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. On the RTX 3060, repeating the experiment with five different random seeds produced the same qualitative family-level behavior.
A separate experiment requested eigenvectors in both paths. After including formation of the Householder matrix and the back-transformation of the eigenvectors, the same qualitative separation remained, although the crossover occurred at larger matrix sizes.
The unequal-variance covariance family behaved in the opposite direction on all three GPUs. Its large initial diagonal spread was associated with substantial transfer of energy away from the diagonal, more Jacobi work, and longer total runtime. Among the tested families, it was also the clear outlier in initial diagonal standard deviation.
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