Polynomial and Rational Measure Modifications of Orthogonal Polynomials via Infinite-Dimensional Banded Matrix Factorizations
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Bibliographic record
Abstract
Abstract We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials. We present several applications and numerical experiments using an open source implementation which make direct use of these results.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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)
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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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