Convergence analysis of optimal SOR for a class of consistently ordered 2-cyclic matrices with complex spectra
Bibliographic record
Abstract
Asymptotic rates of convergence of optimal SOR applied to linear systems with consistently ordered 2-cyclic matrices have been extensively studied in the case where the Jacobi eigenvalues are are real and contained in an interval centered at the origin. It is well known that as the rightmost endpoint of the interval approaches 1 from below, optimal SOR converges an order of magnitude faster than Jacobi. We generalize this to the situation where the Jacobi spectrum is contained in a line segment in the complex plane that is symmetric about the origin. This is an important class of linear systems, which arise often in various physical applications; complex-shifted linear systems are included in this family. Optimal relaxation parameters are known in this case, but a detailed convergence analysis does not seem to exist in the literature. Using techniques of complex analysis, we derive convergence rates, finding that in the complex case they are affected not only by the distance to 1 of the right-hand endpoint of the line segment as in the real case, but also by its phase. We then exploit a useful convergence property that our analysis reveals to generate a hybrid SOR–multigrid scheme and demonstrate its merits as a shifted Laplacian preconditioner for GMRES applied to the Helmholtz equation.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".