Davidson Method

In the Davidson method, an attempt is made to accelerate the convergence of the preconditioned projection Algorithm 11.8 by changing the Rayleigh quotient at each inner iteration step.

For simplicity, we describe here only the most naive version of the Davidson method, without restart, for the pencil $B - \mu A$ (cf. [99,100,335,329,387]):

$$\begin{tabular}{c} $\mu (x^{(i+1)}) = {\rm max}_{ x \in {\ca... ...u^{(i)} A)x^{(i)} \}$. \end{tabular}\vspace{\belowdisplayskip}$$ (286)

Since the dimension of the subspace ${\cal D}_i$ grows with every step, and all basis vectors need to be stored, restarts are usually necessary as in preconditioned projection Algorithm 11.8. A trivial procedure of a restart is to stop the method after a certain number of steps (inner iterations) and then start it again using the last iteration vector as the initial approximation for the next step (of outer iterations).

The method was quite intricate to study theoretically; cf. [88,390]. It is common to use the Davidson method with an indefinite preconditioner (often the diagonal of $B-\mu^{(0)}A$), which further complicates the theoretical analysis.

The Davidson method is popular particularly in the chemistry community (see [232,275]), where the involved matrices are typically diagonally dominant. For such problems, the method often converges quickly even with a simple diagonal preconditioner.