Preconditioning: Difference between revisions
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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>G</math>-vectors | ||
(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} 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 | ||
function which converges to <math>\frac{2m}{\hbar^2 G^2}</math> for large <math>G</math> vectors, and possess | function which converges to <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf[G}</math> vectors, and possess | ||
a constant value for small <math>G</math> vectors. | a constant value for small <math>\mathbf{G}</math> vectors. | ||
We actually use the preconditioning function proposed by Teter et. al{{cite|teter:prb:1989}} | We actually use the preconditioning function proposed by Teter et. al{{cite|teter:prb:1989}} | ||
<math> | <math> | ||
\langle \ | \langle \mathbf{G} | {\bf K} | \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3} | ||
{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad | {27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad | ||
x = \ | x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) }, | ||
</math> | </math> | ||
Revision as of 10:40, 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 $ \epsilon_n$ 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