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

fonte: https://web.archive.org/web/20161121225152/https://en.wikipedia.org/wiki/Folded_spectrum_method

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Folded spectrum method

Literature