Folded spectrum method

Short description: Mathematical method for solving large eigenvalue problems

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

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

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

Literature

• MacDonald, J. K. L. (1934-11-01). "On the Modified Ritz Variation Method". Physical Review (American Physical Society (APS)) 46 (9): 828. doi:10.1103/physrev.46.828. ISSN 0031-899X. Bibcode: 1934PhRv...46..828M. 
• 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. Bibcode: 1994JChPh.100.2394W. 
• https://web.archive.org/web/20070806144253/http://www.sst.nrel.gov/topics/nano/escan.html