Preconditioning: Difference between revisions

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where $ \epsilon_n$ is the exact eigenvalue for the band in interest.
where <math> \epsilon_n</math> is the exact eigenvalue for the band in interest.
Actually the evaluation of this matrix is not possible, recognizing that the
Actually the evaluation of this matrix is not possible, recognizing that the
kinetic energy dominates the Hamiltonian for large <math>G</math>-vectors
kinetic energy dominates the Hamiltonian for large <math>\mathbf{G}</math>-vectors
(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
is  a good idea to approximate the matrix by a diagonal
is  a good idea to approximate the matrix by a diagonal

Revision as of 10:41, 21 March 2019

The idea is to find a matrix which multiplied with the residual vector gives the exact error in the wavefunction. Formally this matrix (the Greens function) can be written down and is given by

where is the exact eigenvalue for the band in interest. Actually the evaluation of this matrix is not possible, recognizing that the kinetic energy dominates the Hamiltonian for large -vectors (i.e. ), it is a good idea to approximate the matrix by a diagonal function which converges to for large Failed to parse (Conversion error. Server ("cli") reported: "[INVALID]"): {\displaystyle \mathbf[G}} vectors, and possess a constant value for small vectors. We actually use the preconditioning function proposed by Teter et. al[1]

with being the kinetic energy of the residual vector. The preconditioned residual vector is then simply

References