Routh–Hurwitz theorem

Let f(z) be a polynomial (with complex coefficients) of degree n with no roots on the imaginary axis (i.e. the line Z = ic where i is the imaginary unit and c is a real number). Let us define ${\displaystyle P_{0}(y)}$ (a polynomial of degree n) and ${\displaystyle P_{1}(y)}$ (a nonzero polynomial of degree strictly less than n) by ${\displaystyle f(iy)=P_{0}(y)+iP_{1}(y)}$, respectively the real and imaginary parts of f on the imaginary line.

Furthermore, let us denote by:

• p the number of roots of f in the left half-plane (taking into account multiplicities);
• q the number of roots of f in the right half-plane (taking into account multiplicities);
• ${\displaystyle \Delta \arg f(iy)}$ the variation of the argument of f(iy) when y runs from −∞ to +∞;
• w(x) is the number of variations of the generalized Sturm chain obtained from ${\displaystyle P_{0}(y)}$ and ${\displaystyle P_{1}(y)}$ by applying the Euclidean algorithm;
• ${\displaystyle I_{-\infty }^{+\infty }r}$ is the Cauchy index of the rational function r over the real line.

With the notations introduced above, the Routh–Hurwitz theorem states that:

${\displaystyle p-q={\frac {1}{\pi }}\Delta \arg f(iy)=\left.{\begin{cases}+I_{-\infty }^{+\infty }{\frac {P_{0}(y)}{P_{1}(y)}}&{\text{for odd degree}}\\[10pt]-I_{-\infty }^{+\infty }{\frac {P_{1}(y)}{P_{0}(y)}}&{\text{for even degree}}\end{cases}}\right\}=w(+\infty )-w(-\infty ).}$

From the first equality we can for instance conclude that when the variation of the argument of f(iy) is positive, then f(z) will have more roots to the left of the imaginary axis than to its right. The equality p − q = w(+∞) − w(−∞) can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is p + q and the w from the right member is the number of variations of a Sturm chain (while w refers to a generalized Sturm chain in the present theorem).

We can easily determine a stability criterion using this theorem as it is trivial that f(z) is Hurwitz-stable iff p − q = n. We thus obtain conditions on the coefficients of f(z) by imposing w(+∞) = n and w(−∞) = 0.

• Plastic number#Geometry

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• Explaining the Routh–Hurwitz Criterion (2020)[1]

• Mathworld entry