Adaptive Eigenvalue Computation - Complexity Estimates
It provides complexity guarantees for adaptive eigenvalue computation, which is important for numerical linear algebra but the results are theoretical and incremental.
The paper designs an adaptive eigenvalue solver for symmetric linear operators that achieves asymptotically optimal complexity by transforming the problem to ℓ₂ space and using gradient-type iterations with dynamically updated error tolerances.
This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on $\ell_2$, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense asymptotically optimal complexity.