Discussing a More Fundamental Concept Than the Minimal Residual Method to Solve Linear System in a Krylov Subspace
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Bibliographic record
Abstract
A more fundamental concept than the minimal residual method is proposed in this paper to solve an $n$-dimensional linear equations system ${\bf A}{\bf x}={\bf b}$ in an $m$-dimensional Krylov subspace. We maximize the orthogonal projection of ${\bf b}$ onto ${\bf y}$: $={\bf A}{\bf x}$. Then, we can prove that the maximal projection solution (MP) is better than that obtained by the least squares solution (LS) with $\|{\bf b}-{\bf A}{\bf x}_{\mbox{\scriptsize MP}}\|<\|{\bf b}-{\bf A}{\bf x}_{\mbox{\scriptsize LS}}\|$. Examples are discussed which confirm the above finding.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.009 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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