Third-order polynomial equation which complex roots

Hi,

I want to plot the three roots of c (real and imaginary) as a function of k for the following third-order polynomial equation:

I am using fsolve to code it but this requires three initial guesses which are hard to identify for the given equation.

Any suggestions?

Thank you

Answer 1

[Carola Forlini]

Thank you for all the answers.

At the end the easiest way was to calculate first the discriminant of thee polynomio for a range of k and then use the roots function to calculate the solution. In this way I have better control on the expected solutions since the discriminant will tell me if I should have all real roots or real and complex conjugates one.

Answer 2

[Dyuman Joshi]

Define the polynomial as a function handle of the variable 'k' and use roots for different values of 'k'.

Also, note that you will need to plot the real and imaginary separately.

Answer 3

[Walter Roberson]

syms c k L
eqn = c^3 ...
    - c^2*(2*k*exp(4*k*L) + 2*k*exp(2*k*L) + exp(4*k*L) - 6*k^2*exp(2*k*L) + 1) / (2*exp(2*k*L) * k * (exp(2*k*L)+1)) ...
    - c*(-k*exp(4*k*L) + 2*k*exp(2*k*L) - k + 2*exp(4*k*L) - 2) / (2*exp(2*k*L)*k^2*(exp(2*k*L) + 1)) ...
    + (exp(4*k*L) - 2*exp(2*k*L) + 1) / (2*exp(2*k*L)*k^3*(exp(2*k*L) + 1))

eqn = 

solutions = solve(eqn, c, 'maxdegree', 3)

solutions = 

sol= subs(solutions, L, 2);   %arbitrary

%vpa(limit(sol(1), k, 0, 'left'))
%vpa(limit(sol(1), k, 0, 'right'))

tiledlayout('flow');
nexttile(); fplot([real(sol(1)), imag(sol(1))], [-3 3]); title('root #1');
nexttile(); fplot([real(sol(2)), imag(sol(3))], [-5 5]); title('root #2');
nexttile(); fplot([real(sol(3)), imag(sol(3))], [-3 3]); title('root #3');

[Walter Roberson]

syms c k L
eqn = c^3 ...
    - c^2*(2*k*exp(4*k*L) + 2*k*exp(2*k*L) + exp(4*k*L) - 6*k^2*exp(2*k*L) + 1) / (2*exp(2*k*L) * k * (exp(2*k*L)+1)) ...
    - c*(-k*exp(4*k*L) + 2*k*exp(2*k*L) - k + 2*exp(4*k*L) - 2) / (2*exp(2*k*L)*k^2*(exp(2*k*L) + 1)) ...
    + (exp(4*k*L) - 2*exp(2*k*L) + 1) / (2*exp(2*k*L)*k^3*(exp(2*k*L) + 1))

eqn = 

solutions = solve(eqn, c, 'maxdegree', 3)

solutions = 

sol= subs(solutions, L, 2);   %arbitrary

%vpa(limit(sol(1), k, 0, 'left'))
%vpa(limit(sol(1), k, 0, 'right'))

tiledlayout('flow');
nexttile(); fplot([real(sol(1)), imag(sol(1))], [-3 -1e-2]); title('root #1');
nexttile(); fplot([real(sol(1)), imag(sol(1))], [1e-2, 3]); title('root #1');

%nexttile(); fplot([real(sol(2)), imag(sol(3))], [-5 5]); title('root #2');
%nexttile(); fplot([real(sol(3)), imag(sol(3))], [-3 3]); title('root #3');

[Walter Roberson]

syms c k L
eqn = c^3 ...
    - c^2*(2*k*exp(4*k*L) + 2*k*exp(2*k*L) + exp(4*k*L) - 6*k^2*exp(2*k*L) + 1) / (2*exp(2*k*L) * k * (exp(2*k*L)+1)) ...
    - c*(-k*exp(4*k*L) + 2*k*exp(2*k*L) - k + 2*exp(4*k*L) - 2) / (2*exp(2*k*L)*k^2*(exp(2*k*L) + 1)) ...
    + (exp(4*k*L) - 2*exp(2*k*L) + 1) / (2*exp(2*k*L)*k^3*(exp(2*k*L) + 1));
solutions = solve(eqn, c, 'maxdegree', 3);
sol= subs(solutions, L, 2);   %arbitrary

%vpa(limit(sol(1), k, 0, 'left'))
%vpa(limit(sol(1), k, 0, 'right'))

tiledlayout('flow');
%nexttile(); fplot([real(sol(1)), imag(sol(1))], [-3 3]); title('root #1');
%nexttile(); fplot([real(sol(2)), imag(sol(3))], [-5 5]); title('root #2');
nexttile(); fplot([real(sol(3)), imag(sol(3))], [-1 1]); title('root #3');

[Walter Roberson]

syms c k L
eqn = c^3 ...
    - c^2*(2*k*exp(4*k*L) + 2*k*exp(2*k*L) + exp(4*k*L) - 6*k^2*exp(2*k*L) + 1) / (2*exp(2*k*L) * k * (exp(2*k*L)+1)) ...
    - c*(-k*exp(4*k*L) + 2*k*exp(2*k*L) - k + 2*exp(4*k*L) - 2) / (2*exp(2*k*L)*k^2*(exp(2*k*L) + 1)) ...
    + (exp(4*k*L) - 2*exp(2*k*L) + 1) / (2*exp(2*k*L)*k^3*(exp(2*k*L) + 1));
solutions = solve(eqn, c, 'maxdegree', 3);
sol= subs(solutions, L, 2);   %arbitrary

vpa(limit(sol(3), k, 0, 'left'))

ans = 

vpa(limit(sol(3), k, 0, 'right'))

ans =