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 ${\displaystyle \varepsilon }$. This means you can get a vector ${\displaystyle \Psi }$ in the middle of the spectrum without solving the matrix.

${\displaystyle \Psi _{i+1}=\Psi _{i}-\alpha (H-\varepsilon \mathbf {1} )^{2}\Psi _{i}}$, with ${\displaystyle 0<\alpha ^{\,}<1}$ and ${\displaystyle \mathbf {1} }$ the Identity matrix.

In contrast to the Conjugate gradient method, here the gradient calculates by twice multiplying matrix ${\displaystyle H:\;G\sim H\rightarrow G\sim H^{2}.}$

Literature

• MacDonald, J. K. L. (1934-11-01). "On the Modified Ritz Variation Method". Physical Review. American Physical Society (APS). 46 (9): 828–828. 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. American Chemical Society (ACS). 98 (8): 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. AIP Publishing. 100 (3): 2394–2397. doi:10.1063/1.466486. ISSN 0021-9606.
• https://web.archive.org/web/20070806144253/http://www.sst.nrel.gov/topics/nano/escan.html