# Folded spectrum method

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 ${\displaystyle \varepsilon }$. This means you can get a vector ${\displaystyle \Psi }$ in the middle of the spectrum without solving the matrix.
${\displaystyle \Psi _{i+1}=\Psi _{i}-\alpha (H-\varepsilon \mathbf {1} )^{2}\Psi _{i}}$, with ${\displaystyle 0<\alpha ^{\,}<1}$ and ${\displaystyle \mathbf {1} }$ the Identity matrix.
In contrast to the Conjugate gradient method, here the gradient calculates by twice multipling matrix ${\displaystyle H:\;G\sim H\rightarrow G\sim H^{2}.}$