Gromov product

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 $\epsilon$. This means you can get a vector $\mathrm{\Psi }$ in the middle of the spectrum without solving the matrix.

${\mathrm{\Psi }}_{i+1}={\mathrm{\Psi }}_i-\alpha (H-\epsilon \mathbf{1}{)}^2{\mathrm{\Psi }}_i$, with $0<{\alpha }^{}<1$ and $\mathbf{1}$ the Identity matrix.

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

Literature

• J. K. L. MacDonald, Phys. Rev. 46, 828 - 828 (1934)
• W. Wang and A. Zunger, J. Phys. Chem. 98, 2158 (1994)
• W. Wang and A. Zunger, J. Chem. Phys. 100, 2394 (1994)
• http://www.sst.nrel.gov/topics/nano/escan.html

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