4. Boundary Locus method: It requires neither the computation of roots of the polynomial nor the solving of simultaneous inequalities. The roots of the stability polynomial are, in general, complex numbers; for the moment let us regard as complex. Then instead of defining an interval of absolute stability to be an interval of the real line such that the roots of lie within the unit circle whenever lies in the interior of the interval, we define a region of absolute stability to be a region of the complex plane such that the roots of lie within the unit circle whenever lies in the interior of the region. Let us call the region R and its boundary . Since the roots of the roots of are continuous functions of , will lie on when all of the roots of lie on the boundary of the unit circle, i.e., when . It follows that the of is given by . For real , the end points of the interval of absolute stability will be given by the points at which cuts the real axis.

Example: Let us illustrate the method for .

For this method,

This is the locus of and it crosses the real axis where or i.e. . At while at The end points of the interval of absolute stability are thus and