Elman, "Convergence analysis of iterative solvers in inexact
Rayleigh Quotient iteration," SIAM Journal on Matrix Analysis and Applications, vol.31, no.3, pp.877-899, 2009.
We can observe that method (b), inexact Rayleigh quotient iteration with a decreasing solve tolerance, achieves the fastest convergence rate with smallest amount of work.
First, we point out the following well-known equivalence between the simplified JacobiDavidson method and Rayleigh quotient iteration for exact system solves, which has been proved in [14, 16, 24], and in [21] for the generalised eigenproblem.
From Lemma 6.2 it is clear that for exact solves one step of simplified Jacobi-Davidson produces an improved approximation to the desired eigenvector that has the same direction as that given by one step of Rayleigh quotient iteration. Hence, as observed in [24], if the correction equation is solved exactly, the method converges as fast as Rayleigh quotient iteration (that is quadratically for nonsymmetric systems).
Figures 6.1 and 6.2 illustrate the convergence history for inexact Rayleigh quotient iteration and simple Jacobi-Davidson.
* If [paralel][r.sup.(i)][paralel]/[absolute value of [[gamma].sup.(i)]] < 1, then there is the potential that one step of the simple inexact Jacobi-Davidson method will perform better than one step of inexact Rayleigh quotient iteration.
* If [paralel][r.sup.(i)][paralel]/[absolute value of [[gamma].sup.(i)]] > 1, then there is the potential that one step of the inexact Rayleigh quotient iteration will perform better than one step of inexact simple Jacobi-Davidson method.
We compare inexact Rayleigh quotient iteration and inexact simple Jacobi-Davidson.
As expected in this case, the convergence rate of inexact Rayleigh quotient iteration is better than the convergence rate of inexact simple Jacobi-Davidson with Rayleigh quotient shift.