Numerical solution of nonlinear matrix equations arising from Green’s function calculations in nano research
Bibliographic record
Abstract
The Green’s function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form X + ( C ∗ + i η D ∗ ) X − 1 ( C + i η D ) = R + i η P , where R and P are Hermitian, P + λ D ∗ + λ − 1 D is positive definite for all λ on the unit circle, and η → 0 + . For each fixed η > 0 , we show that the required solution is the unique stabilizing solution X η . Then X ∗ = lim η → 0 + X η is a particular weakly stabilizing solution of the matrix equation X + C ∗ X − 1 C = R . In nano applications, the matrices R and C are dependent on a parameter, which is the system energy E . In practice one is mainly interested in those values of E for which the equation X + C ∗ X − 1 C = R has no stabilizing solutions or, equivalently, the quadratic matrix polynomial P ( λ ) = λ 2 C ∗ − λ R + C has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute X η efficiently even for very small values of η , thus providing good approximations to X ∗ . We also explain how the solution X ∗ can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of P ( λ ) should be included in the computation. In some applications the matrices C , D , R , P have very special sparsity structures. We show how these special structures can be exploited to drastically reduce the complexity of the doubling algorithm for computing X η .
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 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".