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 make the early parallel Jacobi batches favorable, 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 that Householder preprocessing reduced Jacobi sweep counts in five commonly encountered families. For matrices larger than 32×32 in these families, the reduction in Jacobi runtime exceeded the additional cost of tridiagonalization.
Across these five families and matrix sizes from 64×64 to 1024×1024, the reduction in total runtime ranged from 1.2% to 49.0%.
The contrasting case was an unequal-variance covariance family. Large diagonal spread caused Householder preprocessing to move substantial energy off the diagonal, increasing both Jacobi sweep counts and total runtime. Because variables on very different scales are commonly standardized before analysis, this represents an adverse stress test rather than the usual covariance-data case.
The experiments therefore support the algebraic prediction: diagonal spread is a simple warning sign that Householder preprocessing may be unfavorable.
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