Spectrum folding

The term Folded Spectrum Method (FSM) or spectrum folding describes a mathematical iterative optimization method for eigenvalue problems . With it it is possible to determine the eigenvector of a large eigenvalue problem for the matrix that is closest to any target eigenvalue (from the middle of the spectrum ) without having to solve the entire matrix . ${\displaystyle \Psi }$ ${\displaystyle H}$ ${\displaystyle \varepsilon }$

formula

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

rating

In contrast to the mountain climbing algorithm (English hill climbing , also downhill ) or the CG method , the gradient here (FSM) is determined by applying the matrix twice : ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G\sim H\rightarrow G\sim H^{2}}$

This method is particularly suitable for large, thinly populated matrices .

literature

• JKL MacDonald: On the Modified Ritz Variation Method. In: Physical Review. 46, 1934, p. 828.
• LW Wang, A. Zunger: Electronic Structure Pseudopotential Calculations of Large (.apprx. 1000 Atoms) Si Quantum Dots. In: Journal of Chemical Physics. 98, 1994, p. 2158. doi: 10.1021 / j100059a032
• LW Wang, A. Zunger: Solving Schrodinger's equation around a desired energy: Application to silicon quantum dots. In: Journal of Chemical Physics. 100, 1994, p. 2394.