three order complex coefficient polynomial root matlab

Hi, I would like to find the root for a 3rd order polynomial with complex coefficient.

The polynomial is like:

28x^3 + Ax^2 + Bx - C = 0;

A,B,C are complex numbers.

I appreciate if anyone can help.

Answer 1

[Star Strider]

There are at least two options —

z = complex(randn(3,1), randn(3,1))

z = 
   0.1066 + 0.2511i
   0.8599 + 2.0317i
  -0.2015 + 2.7642i

r = roots([28; z])

r = 
   0.4103 - 0.2811i
  -0.0640 + 0.4828i
  -0.3501 - 0.2106i

syms x 
p = 28*x.^3 + z(1,:)*x.^2 + z(2,:)*x + z(3,:);
vpap = vpa(p, 5)

vpap = 

r = vpa(solve(p), 5)

r = 

.

Answer 2

[John D'Errico]

Another classic solution is to find the matrix that has the same eigenvalues as your polynomial has roots. Then use eig to compute the eigenvalues of this "companion matrix". This is in fact what roots does.

Answer 3

[Walter Roberson]

syms x A B C
eqn = 28*x^3 + A*x^2 + B*x - C == 0;
solutions = solve(eqn, x, 'MaxDegree', 3)

solutions = 

char(solutions(1))

ans = '(C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3) - (B/84 - A^2/7056)/(C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3) - A/84'

char(solutions(2))

ans = '(B/84 - A^2/7056)/(2*(C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3)) - A/84 - (C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3)/2 - (3^(1/2)*((B/84 - A^2/7056)/(C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3) + (C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3))*1i)/2'

char(solutions(3))

ans = '(B/84 - A^2/7056)/(2*(C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3)) - A/84 - (C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3)/2 + (3^(1/2)*((B/84 - A^2/7056)/(C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3) + (C/56 + ((C/56 - A^3/592704 + (A*B)/4704)^2 + (B/84 - A^2/7056)^3)^(1/2) - A^3/592704 + (A*B)/4704)^(1/3))*1i)/2'