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Folded spectrum method

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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 ε {\displaystyle \varepsilon } {\displaystyle \varepsilon }. This means you can get a vector Ψ {\displaystyle \Psi } {\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}} {\displaystyle \Psi _{i+1}=\Psi _{i}-\alpha (H-\varepsilon \mathbf {1} )^{2}\Psi _{i}}, with 0 < α < 1 {\displaystyle 0<\alpha ^{,円}<1} {\displaystyle 0<\alpha ^{,円}<1} and 1 {\displaystyle \mathbf {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}.} {\displaystyle H:\;G\sim H\rightarrow G\sim H^{2}.}

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