finding roots of equation in matlab

Hi. How can I find roots of below equation?

thank you

syms w a b c d
A=a*w^6+b*w^4+c*w^2+d;

Answer 1

[John D'Errico]

In general the roots of a 6th degree polynomial with unknown coefficients cannot be found in an algebraic form. At least not unless you get lucky. Do luck and mathematics seem opposed to you? After all, there are no random aspects of this. :)

Abel-Ruffini proved many years ago that a general 5th degree polynomial or higher

https://en.wikipedia.org/wiki/Abel–Ruffini_theorem

has no algebraic solution in term of radicals. And this is why I said you need something special. Luckily, you really have a cubic polynomial in disguise. And we know how to solve for the roots of a cubic polynomial. At least solve does. I won't claim to have memorized the formula, as it hardly seems worthwhile, even for me.

https://math.vanderbilt.edu/schectex/courses/cubic/

Instead, we can let MATLAB do the heavy lifting.

syms w a b c d
A=a*w^6+b*w^4+c*w^2+d;

syms u
A_u = subs(A,w^2,u)
A_u =
a*u^3 + b*u^2 + c*u + d

usol = solve(A_u,'maxdegree',3)
usol =
                                                                                                                                                                                                                                                                                                                        (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) - b/(3*a) - (- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)
 (- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) - (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2
 (- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) + (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2

wsol = [sqrt(usol);-sqrt(usol)]
wsol =
                                                                                                                                                                                                                                                                                                                         ((((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) - b/(3*a) - (- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))^(1/2)
  ((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) - (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)
  ((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) + (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)
                                                                                                                                                                                                                                                                                                                        -((((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) - b/(3*a) - (- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))^(1/2)
 -((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) - (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)
 -((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) + (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)

As it turns out, solve is smart enough to implicitly figure the same thing out. So I could just have done this:

solve(A,'maxdegree',6)
wsol =
 -((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) - (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)
 -((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) + (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)
                                                                                                                                                                                                                                                                                                                         ((((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) - b/(3*a) - (- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))^(1/2)
                                                                                                                                                                                                                                                                                                                        -((((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) - b/(3*a) - (- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))^(1/2)
  ((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) - (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)
  ((- b^2/(9*a^2) + c/(3*a))/(2*(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)) + (3^(1/2)*((- b^2/(9*a^2) + c/(3*a))/(((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (c/(3*a) - b^2/(9*a^2))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3) + (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3))*1i)/2 - b/(3*a) - (((d/(2*a) + b^3/(27*a^3) - (b*c)/(6*a^2))^2 + (- b^2/(9*a^2) + c/(3*a))^3)^(1/2) - b^3/(27*a^3) - d/(2*a) + (b*c)/(6*a^2))^(1/3)/2)^(1/2)

I have not checked, but my guess is the two set of solutions need not be in the same squence. It looks like they are not. However, they should be the same, after sufficient simplification, and after a proper sorting, none of which I'll do.

[saman ahmadi]

thank you very much for your help dear John D'Errico.

Answer 2

[Image Analyst]

Use roots:

r = roots([a, 0, b, 0, c, 0, d])

Try this:

% A=a*w^6+b*w^4+c*w^2+d;
a = 0.0011
b = 2
c = 3
d = -5000;
w = linspace(-10, 10, 1000);
A=a*w.^6+b*w.^4+c*w.^2+d;
plot(w, A, 'b-', 'LineWidth', 2);
grid on;
xlabel('w', 'FontSize', 18);
ylabel('A', 'FontSize', 18);
yline(0, 'LineWidth', 2); % Draw x axis in black.
r = roots([a, 0, b, 0, c, 0, d])

r =

-3.5527136788005e-15 + 42.6063383539596i

-3.5527136788005e-15 - 42.6063383539596i

-6.9727710817844 + 0i

2.88657986402541e-15 + 7.17643970285413i

2.88657986402541e-15 - 7.17643970285413i

6.9727710817844 + 0i

[saman ahmadi]

I think i didn't explain my aim correctly. I want to find roots this equation(a*w.^6+b*w.^4+c*w.^2+d=0). please guide me. thank you very much