The Routh–Hurwitz stability criterion says that a necessary and sufficient condition for all the roots of the polynomial with real coefficients
to have negative real parts (i.e. is Hurwitz stable) is that
where is the i-th leading principal minor of the Hurwitz matrix associated with .
Using the same notation as above, the Liénard–Chipart criterion is that is Hurwitz stable if and only if any one of the four conditions is satisfied:
- ;\,\Delta _{1}>0,\Delta _{3}>0,\ldots }
- ;\,\Delta _{2}>0,\Delta _{4}>0,\ldots }
- ;\,\Delta _{1}>0,\Delta _{3}>0,\ldots }
- ;\,\Delta _{2}>0,\Delta _{4}>0,\ldots }
Hence one can see that by choosing one of these conditions, the number of determinants required to be evaluated is reduced.
Alternatively Fuller formulated this as follows for (noticing that is never needed to be checked):
;}
This means if n is even, the second line ends in and if n is odd, it ends in and so this is just 1. condition for odd n and 4. condition for even n from above. The first line always ends in , but is also needed for even n.