Circumventing problems introduced by matrix asymmetry in collocation calculations of vibrational spectra by exploiting near symmetry
Bibliographic record
Abstract
Collocation is an enticing alternative to variational methods for solving the vibrational Schr¨odinger equation. It makes it possible to use a general potential without requiring integrals and quadrature. An important disadvantage of collocation is the need to work with nonsymmetric matrices. Eigenvalues of a large matrix are best computed with an iterative method, but iterative eigensolvers are much more efficient for symmetric matrices. Heretofore, it has been costly to use collocation when the basis set and Hamiltonian matrix are large. We demonstrate that it is possible to systematically make the collocation matrix whose eigenvalues one must compute more and more symmetric and propose an efficient iterative eigensolver for a nearly symmetric matrix. Little is known about exploiting near symmetry. We use a combination of filter diagonalization and an iterative linear solver powered by a three-term recursion relation. We test the ideas with a 6-D Hamiltonian and show that accurate energies are obtained despite the asymmetry.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".