Folded spectrum method
In mathematics, the folded spectrum method (FSM) is an iterative method for solving large eigenvalue problems. Here you always find a vector with an eigenvalue close to a search-value ε {\displaystyle \varepsilon } . This means you can get a vector Ψ {\displaystyle \Psi } in the middle of the spectrum without solving the matrix. Ψ i + 1 = Ψ i − α ( H − ε 1 ) 2 Ψ i {\displaystyle \Psi _{i+1}=\Psi _{i}-\alpha (H-\varepsilon \mathbf {1} )^{2}\Psi _{i}} , with 0 < α < 1 {\displaystyle 0<\alpha ^{\,}<1} and 1 {\displaystyle \mathbf {1} } the Identity matrix. In contrast to the Conjugate gradient method, here the gradient calculates by twice multiplying matrix H : G ∼ H → G ∼ H 2 . {\displaystyle H:\;G\sim H\rightarrow G\sim H^{2}.}
In mathematics, the folded spectrum method (FSM) is an iterative method for solving large eigenvalue problems. Here you always find a vector with an eigenvalue close to a search-value . This means you can get a vector in the middle of the spectrum without solving the matrix.
, with and the Identity matrix.
In contrast to the Conjugate gradient method, here the gradient calculates by twice multiplying matrix
Literature
[edit]- MacDonald, J. K. L. (1934-11-01). "On the Modified Ritz Variation Method". Physical Review. 46 (9). American Physical Society (APS): 828. Bibcode:1934PhRv...46..828M. doi:10.1103/physrev.46.828. 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). American Chemical Society (ACS): 2158–2165. doi:10.1021/j100059a032. 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). AIP Publishing: 2394–2397. Bibcode:1994JChPh.100.2394W. doi:10.1063/1.466486. ISSN 0021-9606.
- https://web.archive.org/web/20070806144253/http://www.sst.nrel.gov/topics/nano/escan.html
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