K-Optimal Preconditioners Based on Approximation of Inverse Matrices
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Bibliographic record
Abstract
The problem of constructing preconditioners of a special type for solving systems of linear algebraic equations is considered. A new approach to constructing preconditioners based on minimizing the K-condition number for the matrix $${{A}^{{ - 1}}}P$$ , where $$A$$ is the initial matrix of the system and $$P$$ is the preconditioner, is proposed. It is proved that for circulant matrices such an approach is equivalent to constructing the optimal Chan circulant for the inverse matrix. Numerical experiments are carried out on a series of benchmark problems with Toeplitz matrices, which show that the proposed approach allows one to significantly reduce the number of iterations of the conjugate gradient method compared to the classical approach. The obtained results open up new possibilities for constructing effective preconditioners in other classes of matrices.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 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