Richardson's iteration introduce a scalar $\alpha$ to the update formula:

$$ \textbf{x}^{(k+1)} = \textbf{x}^{(k)} + \alpha \textbf{r}^{(k)} $$

And compute $\alpha$ by minimizing the spectral radius:

$$ min_{\omega}\{\rho(B)\} = min_{\omega}\{\rho(I-\omega A)\} $$

As I,A do not change over the iterations, it might seem it exists only one (i.e. equal for all the iterations) $\omega$ making the max eigenvalue minimal, hence the spectral radius minimal. Since I know gradient method and conjugate gradient perform better by setting $\alpha$ dynamically, I am wondering what am I missing here. Is there any way to see through the spectral radius expression that gradient and conjugate gradient iterations converge faster?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.