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 [31].