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The eigenvalues of the Toeplitz preconditioned Toeplitz Schur complement
Now we want to analyse the spectrum of different preconditioned BT matrices.
If b(x) is real with zeros of even order the spectrum of the preconditioned Schur complement is strongly clustered around 1.
Therefore, the eigenvalues of the preconditioned Schur complement are weakly clustered around 1.
9 has unbounded spectrum of the preconditioned system, e.
2 shows the number of outliers for a Toeplitz preconditioned Toeplitz normal equations.
These methods are usually called preconditioned gradient methods as historically the first method of this kind, suggested in [51], minimizes the Rayleigh quotient on every iteration in the direction of the preconditioned gradient.
Paper [49] obtains asymptotic convergence rate estimate of the generalized Davidson method similar to that by [51] for the preconditioned steepest descent.
Preconditioned gradient methods can be easily linked to the inexact inner-outer iterative solvers discussed earlier in the section.
Preconditioned gradient methods for computing several extreme eigenpairs of symmetric eigenvalue problems have several advantages compared to other preconditioned eigensolvers:
1]B, when a good preconditioner and an advanced preconditioned gradient method, e.
The main drawback of present preconditioned gradient methods is their inability to compute efficiently eigenvalues in a given interval in the middle of the spectrum.