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 $\varepsilon$ . 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 multipling matrix $H:\;G\sim H\rightarrow G\sim H^2.$

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

Numerical linear algebra

Key concepts	Floating point Numerical stability

Problems	Matrix multiplication (algorithms) Matrix decompositions Linear equations Sparse problems

Hardware	CPU cache TLB Cache-oblivious algorithm SIMD Multiprocessing

Software	BLAS Specialized libraries General purpose software

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