<scp>TriCG</scp> and <scp>TriMR</scp>: Two Iterative Methods for Symmetric Quasi-definite Systems
Bibliographic record
Abstract
We introduce iterative methods called TriCG and TriMR for solving symmetric quasi-definite systems based on the orthogonal tridiagonalization process proposed by Saunders, Simon, and Yip in 1988. TriCG and TriMR are tantamount to preconditioned Block-Cg and Block-Minres with two right-hand sides in which the two approximate solutions are summed at each iteration, but require less storage and work per iteration. We evaluate the performance of TriCG and TriMR on linear systems generated from the SuiteSparse Matrix Collection and from discretized and stabilized Stokes equations. We compare TriCG and TriMR with Symmlq and Minres, the recommended Krylov methods for symmetric and indefinite systems. In all our experiments, TriCG and TriMR terminate earlier than Symmlq and Minres on a residual-based stopping condition with an improvement of up to 50% in terms of number of iterations. They also terminate more reliably than Block-Cg and Block-Minres. Experiments in quadruple and octuple precision suggest that loss of orthogonality in the basis vectors is significantly less pronounced in TriCG and TriMR than in Block-Cg and Block-Minres.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.004 |
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".