The rootlocus command plots the complex roots of the equation

as k runs over the range $r=a..b$ . Since $f\left(s\right)=\frac{n\left(s\right)}{d\left(s\right)}$ is a rational function in s, this is equivalent to tracing the paths of the complex roots of the polynomial

Maple's fsolve command is used to compute the roots of the polynomials.  Remaining arguments are interpreted as options which are specified as equations of the form option = value. Standard plot options are supported.

The polynomial is initially solved for n equally spaced points k in the range $r=a..b$. The optional argument $\mathrm{numpoints}=n$ specifies the number of points. The default is 49.

Next, the code tries to pair up the roots for adjacent values of k by choosing those closest to each other. If two sets of roots for $k=\mathrm{k1}$ and $k=\mathrm{k2}$ are not sufficiently distinguishable from one another, the algorithm will compute a new set of roots for $k=\frac{\mathrm{k1}}{2}+\frac{\mathrm{k2}}{2}$. The optional argument adaptive = false will turn this off.

Finally, the code tries to trace out the different curves. It tries to join up adjacent points with line segments. Sometimes this code can be fooled and the result is a messy plot. The optional argument style=point will turn this off by just displaying the roots as points.

The command with(plots,rootlocus) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{plots}\,\mathrm{rootlocus}\right)\:$

$\mathrm{rootlocus}\left(\frac{s^3-1}{s}\,s\,-5..5\right)$

$\mathrm{rootlocus}\left(\frac{s^5-1}{s^2+1}\,s\,-5..5\,\mathrm{style}=\mathrm{point}\,\mathrm{adaptive}=\mathrm{false}\right)$

$\mathrm{rootlocus}\left(\frac{s^5-1}{s^2+1}\,s\,-5..5\,\mathrm{style}=\mathrm{point}\right)$