Therefore dividing an imaginary number by i takes that imaginary number from the imaginary axis
and onto the real axis in exactly the opposite way that multiplying a real number by i takes that real number from the real axis onto the imaginary axis
It is seen that, while the divergence locus is not changed by damping, the Hopf curves, at which a generic dynamic bifurcation occurs, determined by the crossing of the imaginary axis
of one pair of eigenvalues, are affected by the damping parameters.
However, as the derivative of the auxiliary function is positive, eigenvalues always cross imaginary axis
from left to right (independently of n), which means that a switch of stability appears only at [tau] = [[tau].sub.0.
(2) The pole of the controller is kept far from both the zero of the controller and the imaginary axis
Fix a polygon P for the reflection group such that one of the sides is the imaginary axis
[summation] and denote the reflection in [summation] by [sigma].
We remark that whatever the eigenvalues of a matrix are closer to the imaginary axis
, the speed of convergence for different methods becomes slower and more risky to face with singular matrices [X.sub.k], whose inverses could not be computed.
It is clear that convergence will be slow if either [rho](A) [much greater than] 1 or A has eigenvalues close to the imaginary axis
. Hence, it is better to first construct a robust seed by scaled method (9).
According to the scheme of Figure 1, since =1/2, the boundary between the oscillatory behaviour and the nonoscillatory behaviour in the time domain is the imaginary axis
; observing the pole location, it is possible to note that
For obvious reasons the x-axis is called the real axis and the y-axis is called the imaginary axis
where the contour C runs parallel to the imaginary axis
from c - i[infinity] to c + i[infinity] with -[D/2] - 1 < c < -[D/2], and [D/2] the integral part of D/2.
The normal to it is an imaginary axis
, and also an axis of symmetry around which it is possible to combine both quadrants.
Typical applications require a few eigenvalues that are largest or smallest in magnitude or closest to the imaginary axis
. Computing the ones of largest magnitude can be achieved efficiently by Krylov subspace methods, e.g., Arnoldi or Lanczos processes, possibly combined with implicit restarting or a Krylov-Schur-type technique .