$\%$ Matlab code polyplot.m

$\%$ function polyplot$($r$,$e$,$m$)$

$\%$ Form the coefficients of a polynomial specified by its roots,

$\%$ and repeatedly add small perturbations to the coefficients,

$\%$ plotting the resulting perturbed roots

$\%$

$\%$ Inputs:

$\%$ r $=$ vector of polynomial roots

$\%$ e $=$ maximum relative perturbation to make to each coefficient

$\%$ m $=$ number of random polynomials to generate

$\%$

$\%$ Output:

$\%$ Plot of perturbed roots

function polyplot$($r$,$e$,$m$)$

$\%$

$\%$ Generate polynomial coefficients

p$=$poly$($r$)$;

$\%$ Compute min and max roots for plotting

minx $=$ min$($real$($r$))$; maxx $=$ max$($real$($r$))$;

miny $=$ min$($imag$($r$))$; maxy $=$ max$($imag$($r$))$;

$\%$ Set up plot

figure$(1),$ hold off, clf$,$

$\%$ Generate m random polynomials

r$1$save$=[]$;

for i$=1$:m$,$

$\%$ Add random relative perturbation of at most e to each coefficient

p$1=$p$.*($ones$($size$($p$))+$e$*2*($rand$($size$($p$))-\mathrm{.}5))$;

$\%$ Compute and save perturbed roots

r$1=$roots$($p$1)$;

r$1$save$=[$r$1$save;r$1]$;

$\%$ Update min and max for plotting

minx $=$ min$([$real$($r$1)$;minx$])$; maxx $=$ max$([$real$($r$1)$;maxx$])$;

miny $=$ min$([$imag$($r$1)$;miny$])$; maxy $=$ max$([$imag$($r$1)$;maxy$])$;

end

$\%$ If roots all lie in right halfplane, and

$\%$ vary greatly in magnitude, use a semilog plot

if $($minx$>0$ & minx maxx$),$

$\%$ Plot unperturbed and perturbed roots

semilogx$($real$($r$),$imag$($r$),\text{'}$kx$\text{'}),$ hold on

semilogx$($real$($r$1$save$),$imag$($r$1$save$),\text{'}$r$.\text{'})$

else

$\%$ Plot unperturbed and perturbed roots

plot$($real$($r$),$imag$($r$),\text{'}$kx$\text{'}),$ hold on

plot$($real$($r$1$save$),$imag$($r$1$save$),\text{'}$r$.\text{'}),$ hold on$,$

end

if $($ miny $==0$ & maxy $==0)$

axis$([$minx$,$maxx,$-1$e$-300,+1$e$-300])$

elseif $($miny $==$ maxy$)$

axis$([$minx$,$maxx,miny$*\mathrm{.}99,$maxy$*1\mathrm{.}01])$

else

axis$([$minx$,$maxx,miny,maxy$])$

end

grid

title$([\text{'}$Black xs $=$ roots, red dots $=$ perturbed roots'$])$

xlabel$([\text{'}$Max relative perturbation to each coefficient $=\text{'},$num$2$str$($e$)])$