# Folded spectrum method

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Folded_spectrum_method
> Markdown URL: https://mediated.wiki/source/Folded_spectrum_method.md
> Source: https://en.wikipedia.org/wiki/Folded_spectrum_method
> Source revision: 1264242998
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), the **folded spectrum method (FSM)** is an [iterative method](/source/Iterative_method) for solving large [eigenvalue](/source/Eigenvalue) problems. Here you always find a vector with an eigenvalue close to a search-value \varepsilon. This means you can get a vector \Psi in the middle of the spectrum without solving the matrix.

\Psi_{i+1}= \Psi_i-\alpha( H- \varepsilon \mathbf{1} )^2 \Psi_i, with 0<\alpha^{\,}<1 and \mathbf{1} the [Identity matrix](/source/Identity_matrix).

In contrast to the [Conjugate gradient method](/source/Conjugate_gradient_method), here the gradient calculates by twice multiplying matrix H:\;G\sim H\rightarrow G\sim H^2.

## Literature

- MacDonald, J. K. L. (1934-11-01). "On the Modified Ritz Variation Method". *Physical Review*. **46** (9): 828. American Physical Society (APS). [Bibcode:1934PhRv...46..828M](https://ui.adsabs.harvard.edu/abs/1934PhRv...46..828M). [doi:10.1103/physrev.46.828](https://doi.org/10.1103/physrev.46.828). [ISSN 0031-899X](https://www.worldcat.org/issn/0031-899X)
- Wang, Lin Wang & Zunger, Alex (1994). "Electronic Structure Pseudopotential Calculations of Large (.apprx.1000 Atoms) Si Quantum Dots". *The Journal of Physical Chemistry*. **98** (8): 2158–2165. American Chemical Society (ACS). [doi:10.1021/j100059a032](https://doi.org/10.1021/j100059a032). [ISSN 0022-3654](https://www.worldcat.org/issn/0022-3654)
- Wang, Lin-Wang & Zunger, Alex (1994). "Solving Schrödinger's equation around a desired energy: Application to silicon quantum dots". *The Journal of Chemical Physics*. **100** (3): 2394–2397. AIP Publishing. [Bibcode:1994JChPh.100.2394W](https://ui.adsabs.harvard.edu/abs/1994JChPh.100.2394W). [doi:10.1063/1.466486](https://doi.org/10.1063/1.466486). [ISSN 0021-9606](https://www.worldcat.org/issn/0021-9606)
- [https://web.archive.org/web/20070806144253/http://www.sst.nrel.gov/topics/nano/escan.html](https://web.archive.org/web/20070806144253/http://www.sst.nrel.gov/topics/nano/escan.html)

---
Adapted from the Wikipedia article [Folded spectrum method](https://en.wikipedia.org/wiki/Folded_spectrum_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Folded_spectrum_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.