How to Compute Spectra with Error Control

Colbrook, Matthew J.; Roman, Bogdan; Hansen, Anders C.

doi:10.1103/physrevlett.122.250201

Cited by 59 publications

(59 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We finally note that recent years have seen a flurry of activity in this direction with many results classifying various problems within the SCI Hierarchy. We point out [11,12] where some of the theory of spectral computations has been further developed; [32] where this has been applied to certain classes of unbounded operators; [2] where solutions of PDEs were considered; [6,7] where we considered resonance problems; and [13] where the authors give further examples of how to perform certain spectral computations with error bounds.…”

Section: Previous Resultsmentioning

confidence: 99%

Universal algorithms for computing spectra of periodic operators

2022

View full text Add to dashboard Cite

Schrödinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a ‘one-size-fits-all’ algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schrödinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schrödinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.

show abstract

Section: Previous Resultsmentioning

confidence: 99%

Universal algorithms for computing spectra of periodic operators

2022

View full text Add to dashboard Cite

show abstract

“…Separately, recent years have seen a flurry of activity in research revolving around the SCI concept. In addition to [6,7,20] which have been mentioned above, we point out [12,13] where some of the theory of spectral computations has been further developed; [32] where this has been applied to certain classes of unbounded operators; [4] where solutions of PDEs were considered; and [14] where the authors show how to perform certain spectral computations with error bounds.…”

Section: Discussionmentioning

confidence: 99%

Computing the Sound of the Sea in a Seashell

2021

View full text Add to dashboard Cite

The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is $${{\mathcal {C}}}^2$$ C 2 . The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.

show abstract

“…Rather then working with square sections of the infinite matrix T , one should work with uneven sections P n T P m , where the parameters n and m are allowed to vary independently. Indeed, the algorithms presented in [24,38] use this method. In effect, we need to know how large n should be to retain enough information of the operator T P m .…”

Section: Remark 63mentioning

confidence: 99%

“…Note that the SCI hierarchy can be refined. We will not consider the full generalisation in the higher part of the hierarchy in this paper, but recall the class 1 [24]. This class is defined as follows.…”

mentioning

confidence: 99%

On the infinite-dimensional QR algorithm

Colbrook

Hansen

2019

Numer. Math.

Self Cite

View full text Add to dashboard Cite

Spectral computations of infinite-dimensional operators are notoriously difficult, yet ubiquitous in the sciences. Indeed, despite more than half a century of research, it is still unknown which classes of operators allow for the computation of spectra and eigenvectors with convergence rates and error control. Recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation, and no convergence rates nor error control is possible. This begs the question: which classes of operators allow for computations with convergence rates and error control? In this paper, we address this basic question, and the algorithm used is an infinitedimensional version of the QR algorithm. Indeed, we generalise the QR algorithm to infinite-dimensional operators. We prove that not only is the algorithm executable on a finite machine, but one can also recover the extremal parts of the spectrum and corresponding eigenvectors, with convergence rates and error control. This allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture. The algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches (that are notorious for providing false solutions). We also find that in some cases the IQR algorithm performs better than predicted by theory and make conjectures for future study.

show abstract

How to Compute Spectra with Error Control

Cited by 59 publications

References 66 publications

Universal algorithms for computing spectra of periodic operators

Universal algorithms for computing spectra of periodic operators

Computing the Sound of the Sea in a Seashell

On the infinite-dimensional QR algorithm

Contact Info

Product

Resources

About