A note on the Routh-Hurwitz test

This note demonstrates how the ε-method in the original Routh-Hurwitz test can be applied when the Routh array contains a row with zero leftmost element together with an all-zero row. The complete root distribution is determined by applying the criterion only once to the given polynomial without either factoring out a common divisor or shifting the imaginary axis. Since the test may become computationally tedious due to the inclusion of cumbersome ε-terms, the recently reported method of Shamash [1] is used to simplify the computational scheme. This method, being restricted to the case where has purely imaginary roots constituting the whole set of roots of the greatest common even divisor, is here refined and generalized to accommodate all cases in which all-zero rows appear.