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 $\varepsilon$ . 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.

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

Literature

{{cite journal | last=MacDonald | first=J. K. L. | title=On the Modified Ritz Variation Method | journal=Physical Review | publisher=American Physical Society (APS) | volume=46 | issue=9 | date=1934-11-01 | issn=0031-899X | doi=10.1103/physrev.46.828 | pages=828| bibcode=1934PhRv...46..828M }}
{{cite journal | last1=Wang | first1=Lin Wang | last2=Zunger | first2=Alex | title=Electronic Structure Pseudopotential Calculations of Large (.apprx.1000 Atoms) Si Quantum Dots | journal=The Journal of Physical Chemistry | publisher=American Chemical Society (ACS) | volume=98 | issue=8 | year=1994 | issn=0022-3654 | doi=10.1021/j100059a032 | pages=2158–2165}}
{{cite journal | last1=Wang | first1=Lin-Wang | last2=Zunger | first2=Alex | title=Solving Schrödinger's equation around a desired energy: Application to silicon quantum dots | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=100 | issue=3 | year=1994 | issn=0021-9606 | doi=10.1063/1.466486 | pages=2394–2397| bibcode=1994JChPh.100.2394W }}
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Category:Numerical linear algebra