How to Compute Spectra with Error Control
| Autor: | Anders C. Hansen, Matthew J. Colbrook, Bogdan Roman |
|---|---|
| Přispěvatelé: | Colbrook, Matthew [0000-0003-4964-9575], Apollo - University of Cambridge Repository |
| Rok vydání: | 2019 |
| Předmět: |
Phase transition
Open problem Spectrum (functional analysis) General Physics and Astronomy Boundary (topology) Statistical mechanics 01 natural sciences Power (physics) 0103 physical sciences 0202 Atomic Molecular Nuclear Particle and Plasma Physics 010306 general physics Error detection and correction Algorithm Eigenvalues and eigenvectors |
| DOI: | 10.17863/cam.42920 |
| Popis: | Computing the spectra of operators is a fundamental problem in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and the algorithms provided are optimal, realizing the boundary of what digital computers can achieve. Moreover, they are easy to implement and parallelize, offer fundamental speed-ups, and allow problems that before, regardless of computing power, were out of reach. Results are demonstrated on difficult problems such as the spectra of quasicrystals and non-Hermitian phase transitions in optics. |
| Databáze: | OpenAIRE |
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