GPMR: An Iterative Method for Unsymmetric Partitioned Linear Systems
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Bibliographic record
Abstract
We introduce an iterative method named Gpmr (general partitioned minimum residual) for solving block unsymmetric linear systems. Gpmr is based on a new process that simultaneously reduces two rectangular matrices to upper Hessenberg form and is closely related to the block-Arnoldi process. Gpmr is tantamount to Block-Gmres with two right-hand sides in which the two approximate solutions are summed at each iteration, but its storage and work per iteration are similar to those of Gmres. We compare the performance of Gpmr with Gmres on linear systems from the SuiteSparse Matrix Collection. In our experiments, Gpmr terminates significantly earlier than Gmres on a residual-based stopping condition with an improvement ranging from around 10% up to 50% in terms of number of iterations.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it