Third-order polynomial equation which complex roots

2024 2 3
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更新时间：2024 2 6
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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

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Carola Forlini
Carola Forlini 2024-2-6

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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.

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Dyuman Joshi
Dyuman Joshi 2024-2-3
移动：Matt J 2024-2-3

0 个投票

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.
Walter Roberson
Walter Roberson 2024-2-3
Ran in:

0 个投票

Ran in:
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');

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Walter Roberson
Walter Roberson 2024-2-3
Ran in:
Ran in:
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
Walter Roberson 2024-2-3
Ran in:
Ran in:
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
Walter Roberson 2024-2-3
Ran in:
Ran in:
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 = 

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