Numerical and symbolic computation of the Lambert W function in C <sup>nxn</sup>
Bibliographic record
Abstract
The Lambert W function is a multivalued complex function, first named in the computer algebra system Maple. We present iterative schemes and strategies for the numerical evaluation of all branches of the scalar complex Lambert W function to hardware precision with high computational efficiency, and present a set of rules for the simplification of special symbolic arguments. We also extend the numerical and symbolic computations to the Lambert W function in C nxn , for n > 1. In order to achieve high precision and computational efficiency, we evaluate a series of high order and classical iterative methods and strategies for the evaluation of the scalar Lambert W function. We then construct optimal iterative schemes for the evaluation of the complex Lambert W function in the IEEE oating point model. The schemes consist of variations on Newton and Halley iterations together with initial estimates generated using a variety of series approximations. We also study several classes of exact simplifications for the Lambert W function for symbolic arguments and give rules for their application. Finally, we consider the solutions of the matrix equation S exp(S) = A, where S and A are n x n matrices. The solutions are expressed in terms of extensions of the scalar Lambert W function to C nxn . The solutions of the matrix equations consist not only of the matrix functions W(A); other solutions also exist. We focus first on solving the matrix equation in C 3x3 , and implement solutions in the floating-point case, and the symbolic case, using Maple.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.003 |
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".