Routh Zero Location Tests Unhampered by Nonessential Singularities

The generalization of the Routh stability test to the corresponding zero location problem of determining how many of the zeros of a polynomial have negative real parts, positive real parts, or are purely imaginary encountered intensive and controversial activity about overcoming singular cases. This paper revisits this problem and presents two (MaxQ and LinQ) ZL methods for an arbitrary complex polynomial. The first method produces for an investigated polynomial
$P(s)$
of degree
$n$
a sequence of length
$\leq n+1$
of para-even or para-odd (para-paritic) polynomials of descending degrees obtained by a polynomial remainder routine with quotients of
maximal
degrees. The second method uses successively
linear
quotients for the reduction of degrees and thus assign to
$P(s)$
always a sequence of exactly
$n+1$
para-paritic polynomials. Previous so-called first-type singularities become “non-essential singularities” in the sense that they are now absorbed into modified forms of the polynomial recursions. The distribution of zeros with respect to the imaginary axis is extracted in both methods by sign variation rules posed on the leading coefficients (that stay real when testing a complex polynomial as well) of the polynomials in the sequence.