Folded spectrum method

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the folded spectrum method (FSM) is an <a href="/facts/Iterative_method/uKMAJhEh">iterative method</a> for solving large <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> problems.
Here you always find a vector with an eigenvalue close to a search-value 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
. This means you can get a vector 
 
 
 
 Ψ
 
 
 {\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}}
 
, with 
 
 
 
 0
 <
 
 α
 
 
 
 
 <
 1
 
 
 {\displaystyle 0<\alpha ^{\,}<1}
 
 and 
 
 
 
 
 1
 
 
 
 {\displaystyle \mathbf {1} }
 
 the <a href="/facts/Identity_matrix/Glbcf6ol">Identity matrix</a>.
In contrast to the <a href="/facts/Conjugate_gradient_method/AFwzBS4U">Conjugate gradient method</a>, here the gradient calculates by twice multiplying matrix 
 
 
 
 H
 :
 
 G
 ∼
 H
 →
 G
 ∼
 
 H
 
 2
 
 
 .
 
 
 {\displaystyle H:\;G\sim H\rightarrow G\sim H^{2}.}

Folded spectrum method open-in-new

Folded spectrum method