How to plot and find zeros of a complex polynomial?

Given any complex polynomial, how to find and plot zeros in a complex plane? Here are a few examples what kind of plot i want...

Answer 1

[Sam Chak]

Hi @simran

I am unsure whether a symbolic approach exists. However, to use the fsolve() function, you need to provide initial guesses, which I have roughly estimated based on your image.

% Complex polynomial
function F = complexPolynomial(z)
    F(1) = z(1)^5 + 1.2*conj(z(1))^4 - 1;   % Real part
    F(2) = imag(z(1));                      % Imaginary part
end

% Initial guesses for z
initial_guesses = [1.0 + 0.0i; 
                  -1.0 - 1.0i; 
                  -1.0 + 1.0i; 
                   0.6 - 1.2i; 
                   0.6 + 1.2i; 
                   1.0 - 0.3i; 
                   1.0 + 0.3i];

% For storing solutions
solutions = [];

% Solve using fsolve for each initial guess
options = optimoptions('fsolve', 'Algorithm', 'levenberg-marquardt', 'Display', 'off'); 
for i = 1:length(initial_guesses)
    z0       = initial_guesses(i);
    solution = fsolve(@complexPolynomial, z0, options);
    
    % Check if the solution is unique
    if all(abs(solutions - solution(1)) >= 1e-6) % Tolerance for uniqueness
        solutions = [solutions; solution(1)];
    end
end

disp('Solutions:');

Solutions:

disp(solutions);

   0.8370 + 0.0000i
  -1.0995 - 0.9097i
  -1.0995 + 0.9097i
   0.6501 - 1.1865i
   0.6501 + 1.1865i
   1.0506 - 0.3025i
   1.0506 + 0.3025i

% Plot 
G   = zpk([], solutions, 1);
pzmap(G), grid on