Curves of roots of rational expressions
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plot::Rootlocus(p(z, u)
, u = u_{min} .. u_{max}
, <a = a_{min} .. a_{max}
>, options
)
plot::Rootlocus(p(z, u), u = u_{min}..
u_{max})
creates a 2D plot of the curves
in the complex plane given by the roots of p(z, u)
= 0 (solved for z)
as the parameter u varies
between u_{min}
and u_{max}
.
For any given value of u, plot::Rootlocus
solves
the equation p(z, u)
= 0 for z.
The solutions define points with coordinates x = ℜ(z), y = ℑ(z) in
the complex plane. As the parameter u varies,
the solutions form continuous curves that a depicted by plot::Rootlocus
.
The roots of the numerator of p(z, u)
are
considered. All complex solutions of this polynomial in z are
computed numerically via numeric::polyroots
.
The polynomial is initially solved for some values u from
the range u = `u_{min}` .. `u_{max}`
. The optional
argument Mesh
= n can
be used to specify the number n of
these initial points (the default value is 51). These points are not
equally spaced, but accumulate close to the end of the range.
The routine then tries to pair up the roots for adjacent values of u by choosing those closest to each other.
Finally, the routine tries to trace out the different curves
by joining up adjacent points with line segments. If adjacent line
segments exhibit angles that are not close to 180 degrees, additional
roots are computed for parameter values u between
the values of the initial mesh. Up to m such
bisectioning steps are possible, where m is
specified by the optional argument AdaptiveMesh
= m (the
default value is 4). With AdaptiveMesh
= 0,
this adaptive mechanism may be switched off.
Sometimes, the matching up of the roots to continuous curves can be fooled and the result is a messy plot. In such a case, the user can take the following measures to improve the plot:
The parameter range u = `u_{min}` .. `u_{max}`
may
be unreasonably large. Reduce this range to a reasonable size!
Increase the size n of
the initial mesh using the option Mesh
= n.
Note that increasing n by
some factor may increase the runtime of the plot by the same factor!
Increase the number m of
possible adaptive bisectioning steps using the option AdaptiveMesh
= m.
Note that increasing m by 1 may
increase the runtime of the plot by a factor of 2!
Using the options LinesVisible = FALSE
in
conjunction with PointsVisible = TRUE
, the roots
are displayed as separate points without joining line segments.
Cf. Example 2.
Animations are triggered by specifying a range a =
`a_{min}` .. `a_{max}`
for a parameter a
that
is different from the variables z
and u
.
Cf. Example 3.
The curves can be colored by a user defined color scheme. Just
pass the option LineColorFunction
= mycolor
,
where mycolor
is a user definied procedure that
returns an RGB
color value. The routine plot::Rootlocus
calls mycolor(u,
x, y)
, where u is
the parameter value and x, y are
the real and imaginary parts of the root of p(x +
i y, u) = 0.
Cf. Example 4.
Attribute  Purpose  Default Value 

AdaptiveMesh  adaptive sampling  4 
AffectViewingBox  influence of objects on the ViewingBox of
a scene  TRUE 
AntiAliased  antialiased lines and points?  TRUE 
Frames  the number of frames in an animation  50 
Legend  makes a legend entry  
LegendText  short explanatory text for legend  
LegendEntry  add this object to the legend?  FALSE 
LineColor  color of lines  RGB::Blue 
LineWidth  width of lines  0.35 
LineStyle  solid, dashed or dotted lines?  Solid 
LinesVisible  visibility of lines  TRUE 
LineColorType  line coloring types  Flat 
LineColorFunction  functional line coloring  
Mesh  number of sample points  51 
Name  the name of a plot object (for browser and legend)  
ParameterEnd  end value of the animation parameter  
ParameterName  name of the animation parameter  
ParameterBegin  initial value of the animation parameter  
ParameterRange  range of the animation parameter  
PointSize  the size of points  1.0 
PointStyle  the presentation style of points  FilledCircles 
PointsVisible  visibility of mesh points  FALSE 
RationalExpression  rational expression in a rootlocus plot  
TimeEnd  end time of the animation  10.0 
TimeBegin  start time of the animation  0.0 
TimeRange  the real time span of an animation  0.0 .. 10.0 
Title  object title  
TitleFont  font of object titles  [" sansserif " , 11 ] 
TitlePosition  position of object titles  
TitleAlignment  horizontal alignment of titles w.r.t. their coordinates  Center 
TitlePositionX  position of object titles, x component  
TitlePositionY  position of object titles, y component  
UMax  final value of parameter "u"  
UMesh  number of sample points for parameter "u"  51 
UMin  initial value of parameter "u"  
UName  name of parameter "u"  
URange  range of parameter "u"  
Visible  visibility  TRUE 
VisibleAfter  object visible after this time value  
VisibleBefore  object visible until this time value  
VisibleFromTo  object visible during this time range  
VisibleAfterEnd  object visible after its animation time ended?  TRUE 
VisibleBeforeBegin  object visible before its animation time starts?  TRUE 
The roots of the polynomial z^{2}  2 u z + 1 are given by and . We visualise these two curves via a rootlocus plot:
plot(plot::Rootlocus(z^2  2*u*z + 1, u = 1.5..1.5))
For rational expressions, the roots of the numerator are considered. The following plot displays the roots of the numerator polynomial (z^{2}  u)^{2} + u (z  u)^{3}:
plot(plot::Rootlocus(1 + u * (z  u)^3/(z^2  u)^2, u = 1..1)):
Here are various other examples:
plot(plot::Rootlocus((z^2  2*u*z + 1)^2 + u, u = 1..1))
plot(plot::Rootlocus((z^2  u)^6 + u^2, u = 2..2, Color = RGB::Red))
plot(plot::Rootlocus((z^5  1)^3 + u, u = 1..1, PointsVisible, PointSize = 1.5))
The following plot is rather messy, since the default mesh size of 51 initial points on each curve is not sufficient to obtain a good resolution:
plot(plot::Rootlocus((zu)^3  u/z^3, u = 10^3 .. 10^3)):
We obtain a better resolution by decreasing the range of the parameter u to a reasonable size. There are still a few points that are not properly matched up with the curves:
plot(plot::Rootlocus((zu)^3  u/z^3, u = 10 .. 10)):
We increase the mesh size to cure this problem:
plot(plot::Rootlocus((zu)^3  u/z^3, u = 10 .. 10, Mesh = 251)):
We plot the roots as separate points without displaying connecting line segments:
plot(plot::Rootlocus((zu)^3  u/z^3, u = 10 .. 10, Mesh = 501, LinesVisible = FALSE, PointsVisible)):
We animate the expression whose roots are to plotted:
plot(plot::Rootlocus(z^2  2*u*z + a, u = 1..1, a = 0.2 .. 2, Mesh = 10), plot::Text2d(a > "a = ".stringlib::formatf(a, 2, 5), [1.2, 1.0], a = 0.2 .. 1));
We animate the parameter range:
plot(plot::Rootlocus(z^2  2*u*z + 0.81, u = 1 .. a, a = 1 .. 1, Mesh = 10))
We provide a color function: roots for small values of the parameter u are displayed in red, whereas roots for large parameter values are displayed in blue:
plot(plot::Rootlocus(z^2  2*u*z + 0.81, u = 1..1, LineColorFunction = ((u, x, y) > [(1  u)/2, 0, (1 + u)/2])))

An arithmetical expression in
two unknowns z and u and,
possibly, the animation parameter


Name of the unknown: an identifier or an indexed identifier. 

Name of the curve parameter: an identifier or an indexed identifier.


The range of the curve parameter:


Animation parameter, specified as 