three order complex coefficient polynomial root matlab

3 views (last 30 days)
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.

Answers (3)

Star Strider
Star Strider on 26 May 2023
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 on 26 May 2023
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 on 26 May 2023
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'

Categories

Find more on Polynomials in Help Center and File Exchange

Products

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!