<scp>TriCG</scp> and <scp>TriMR</scp>: Two Iterative Methods for Symmetric Quasi-definite Systems
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.013 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.005 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.007 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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 it