Computing an Optimal Ellipse

We would like to find the ‘best’ ellipse enclosing the setRof unwanted eigenval- ues, i.e., the eigenvalues other than the ones with theralgebraically largest real parts. We must begin by clarifying what is meant by ‘best’ in the present context. Consider Figure 7.3 representing a spectrum of some matrixAand suppose that we are interested in therrightmost eigenvalues, i.e.,r= 4in the figure.

ℜe(z) ℑm(z) - 6 * * * * * * * * * * * * λ1 λ2 λ3 λ4

Figure 7.3: Example of a spectrum and the enclosing best ellipse forr= 4. Ifr = 1then one may simply seek the best ellipse in the sense of minimiz- ing the convergence ratioτ(λ1). This situation is identical to that of Chebyshev Iteration for linear systems for which much work has been done.

ℜe(z) ℑm(z) - 6 * * * * * * * λ1 λ2 λ3 λ4 λ5 λn−1 λn

Figure 7.4: Case whereµ=λr(complex): the eigenvaluesλ2andλ3are inside the ‘best’ ellipse.

Whenr > 1,then we have several convergence ratios, each corresponding to one of the desired eigenvaluesλi, i= 1,· · · , r, and several possible strategies may be defined to try to optimize the process.

Initially, assume thatλris real (Figure 7.3) and consider any ellipseE(c, e, a)

including the setRof unwanted eigenvalues and not the eigenvalues

{λ1, λ2,· · ·, λr}.

It is easily seen from our comments of subsection 7.4.1 that if we draw a vertical line passing through the eigenvalue λr, all eigenvalues to the right of the line will converge faster than those to the left. Therefore, whenλr is real, we may simply define the ellipse as the one that minimizes the convergence ratio ofλr

with respect to the two parameterscande.

Whenλris not real, the situation is more complicated. We could still attempt to maximize the convergence ratio for the eigenvalueλr,but the formulas giving the optimal ellipse do not readily extend to the case whereλr is complex and the best ellipse becomes difficult to determine. But this is not the main reason why this choice is not suitable. A close look at Figure 7.3, in which we assume

r= 5,reveals that the best ellipse forλrmay not be a good ellipse for some of the desired eigenvalues. For example, in the figure the eigenvaluesλ2, λ3should be computed before the pairλ4, λ5since their real parts are larger. However, because they are enclosed by the best ellipse forλ5they may not converge until many other eigenvalues will have converged including λ4, λ5, λn, λn−1 and possibly other

The difficulty comes from the fact that this strategy will not favor the eigen- values with largest real parts but those belonging to the outermost confocal el- lipse. It can be resolved by just maximizing the convergence ratio ofλ2 instead ofλ5in this case. In a more complex situation it is unfortunately more difficult to determine at which particular eigenvalueλkor more generally at which value

µit is best to maximize τ(µ). Clearly, one could solve the problem by taking

µ=ℜe(λr),but this is likely to result in a suboptimal choice.

As an alternative, we can take advantage of the previous ellipse, i.e., an ellipse determined from previous purification steps. We determine a pointµon the real line having the same convergence ratio asλr,with respect to the previous ellipse. The next ‘best’ ellipse is then determined so as to maximize the convergence ratio for this point µ. This reduces to the previous choice µ = ℜe(λr) whenλr is real. At the very first iteration one can setµto be_ℜe(λr). This is illustrated in Figure7.5. In Figure 7.5 the ellipse in solid is the optimal ellipse obtained from some previous calculation from the dynamic process. In dashed line is an ellipse that is confocal to the previous ellipse which passes throughλr. The pointµis defined as one of the two points where this ellipse crosses the real axis.

ℜe(z) ℑm(z) - 6 ←Previous ellipse ↑

Ellipse of the same family *

*

λr

µ

Figure 7.5: Point on the real axis whose convergence is equivalent with that ofλr

with respect to the previous ellipse.

The question which we have not yet fully answered concerns the practical determination of the best ellipse. At a typical step of the Arnoldi process we are givenmapproximationsλi, i˜ = 1,· · · , m,of the eigenvalues ofA. This approx- imate spectrum is divided in two parts: therwanted eigenvalues˜λ1,· · ·,λr˜ and the setR˜of the remaining eigenvaluesR˜={λr+1,˜ · · ·,λm}˜ . From the previous ellipse and the previous setsR,˜ we would like to determine the next estimates for the optimal parameterscande.

A similar problem was solved in the context of linear systems of equations and the technique can easily be adapted to our situation. We refer the reader to the two articles by Manteuffel [126, 127]. The change of variablesξ = (µ−λ)

easily transformsµinto the origin in theξ–plane and the problem of maximizing the ratioτ(µ)is transformed into one of maximizing a similar ratio in theξ–plane for the origin, with respect to the parameters c ande. An effective technique for solving this problem has been developed in [125], [127] but its description is rather tedious and will be omitted. We only indicate that there exist reliable software that will deliver the optimal values ofµ−candeat output if given the shifted eigenvaluesµ−λj,˜ j=r+ 1,· · ·, mon input.

We now wish to deal with a minor difficulty encountered whenλ1is complex. Indeed, it was mentioned in Section 7.4 that the eigenvalueλ1in (7.13) should, in practice, be replaced by some approximationνofλ1. Initially,ν can be set to some initial guess. Then, when the approximation˜λ1as computed from the outer loop of the procedure, becomes available it can be used. If it is real then we can takeν = ˜λ1and the iteration can be carried out in real arithmetic as was already shown, even whene is purely imaginary. However, the iteration will become complex if˜λ1is complex. To avoid this it suffices to takeν to be one of the two points where the ellipseE(c, e, a1)passing throughλ1˜ , crosses the real axis. The effect of the corresponding scaling of the Chebyshev polynomial will be identical with that using˜λ1but will present the advantage of avoiding complex arithmetic.