H-Matrix approximation for the operator exponential with applications
H-Matrix approximation for the operator exponential with applications
H-Matrix approximation for the operator exponential with applications
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H-<strong>Matrix</strong> <strong>approximation</strong> <strong>for</strong> <strong>the</strong> <strong>operator</strong> <strong>exponential</strong> <strong>with</strong> <strong>applications</strong> 109<br />
<strong>the</strong>n<br />
{<br />
Γ b(k) (−d) = z = a k η2 + b + k − 1<br />
4a<br />
+ i η<br />
{<br />
(6.17) = z = aη∗ 2 + γ 0 + iη ∗ : η ∗ ≡<br />
≡ Γ 0 .<br />
}<br />
√ : η ∈ (−∞, ∞)<br />
k<br />
}<br />
η √<br />
k<br />
∈ (−∞, ∞)<br />
From (6.15), one can see that a ′ → 0, b ′ → 0 monotonically <strong>with</strong>respect<br />
to ν as ν →∞, i.e. <strong>the</strong> parabolae from Γ b (ν) move away from <strong>the</strong> spectral<br />
parabola Γ 0 monotonically. This means that <strong>the</strong> parabolae set Γ b (ν) <strong>for</strong><br />
b = b(k), |ν| 0.<br />
and<br />
We have also<br />
(6.20) ‖F (η, t; L)‖