How can I determine which roots are closest to the unit circle?

Hi all!

I have a problem that I'm currently facing. Lets say I have a column vector (6x1) in which all elements provides me the roots of a polynomial.

For example,

root_out = 6×1 complex
 -2.446362435057714 - 0.000000000000002i
 -0.408770174717145 - 0.000000000000000i
  0.862312857431274 - 0.529586047198464i
  0.842065522070228 - 0.517151225883085i
  0.862312857431279 + 0.529586047198471i
  0.842065522070224 + 0.517151225883078i 

Out of this list (Which may vary in size for its rows), how do I determine which root_out elements (could be more than 1) are closest to the unit circle efficiently? (Maybe in something like a search/range function?)

Answer 1

[Jan]

x = [ -2.446362435057714 - 0.000000000000002i, ...
 -0.408770174717145 - 0.000000000000000i, ...
  0.862312857431274 - 0.529586047198464i, ...
  0.842065522070228 - 0.517151225883085i, ...
  0.862312857431279 + 0.529586047198471i, ...
  0.842065522070224 + 0.517151225883078i, ...
  0.842065522070228 - 0.517151225883085i];  % Repeated 4th line
r = abs(x);
d = abs(r - 1);
index  = (d == min(d));
result = x(index)

Now result contains the value(s), which are nearest to the unit circle. I've repeated the 4th value to test the output of multiple minimal values.

[Murdifi Bin Muhammad]

Hi Jan!

This worked exceptionally well! But how do I modify the code in such a way where I would like N-number of elements closest to the unit circle instead of just 1 element?

Thank you so much for the assistance in advance!

Murdifi

[Jan]

Find n=3 points with smallest distance to unit circle:

n      = 3;
radius = abs(x);
dist   = abs(r - 1);
[~, index] = sort(dist);
x(index(1:n))

With Matlab >= R2017b:

n      = 3;
radius = abs(x);
dist   = abs(r - 1);
[~, index] = mink(dist, n);
x(index)

[Murdifi Bin Muhammad]

Thanks Jan! This worked well! You're a lifesaver!

Murdifi

Answer 2

[Torsten]

[~,ix] = min(abs(real(root_out).^2+imag(real_out).^2-1)./sqrt(real(root_out).^2+imag(real_out).^2));
root_out(ix)