three order complex coefficient polynomial root matlab

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Alex Lee
Alex Lee 2023-5-26
回答： Walter Roberson 2023-5-26
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.

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回答（3 个）

Star Strider
Star Strider 2023-5-26
Ran in:
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 = 
.
John D'Errico
John D'Errico 2023-5-26
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.
Walter Roberson
Walter Roberson 2023-5-26
Ran in:
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'

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