Consider linear equation $Au = f$.

We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f) - \beta(u^k - u^{k-1}). $$

For both methods we can define iteration parameters $\alpha_k$, $\beta_k$ via minimax problem which solution is Chebyshev polynomials.

This is good, but it seems to me, that convergence speed is the same for both cases and is $$ \|\varepsilon^{k}\| \leq \frac{2\sigma^k}{1+\sigma^{2k}}\|\varepsilon^{0}\|, $$
where $u - u^k = \varepsilon^{k}$ approximation error after the $k$-th iteration and $\sigma$ is constant dependent on operator spectrum.

The only idea I have, that first order iterations optimal for chosen $k$ for which coefficients are computed, while second-order iteration is optimal on every step.

I would greatly appreciate any work on this to clear-up those details.


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