I get "Warning: Unable to determine poles" when I try to find the poles and zeros of a 5th-order equation with multiple variables.

I am trying to find the poles and zeros of an equation with 6 variables other than s. If I assign numbers to all of the variables, I get numerical poles and zeros, but what I want is poles and zeros as an expression of the variables. When I try that, it responds, "Warning: Unable to determine poles." I tried adding some reasonable assumptions but got the same answer.

Here is the code that works:

>> clear;

syms R A B C M N s;

assume(in(R , 'real') & R>0);

assume(in(A , 'real') & A>0 & A<1);

assume(in(B , 'real') & B>0 & B<1);

assume(in(C , 'real') & C>0 & C<1);

assume(in(M , 'real') & M>0 & M<1);

assume(in(N , 'real') & N>0 & N<1);

R=13;

A=1/2;

B=1/3;

C=1/5;

M=1/7;

N=1/11;

H = R/(1+R*(A+B+C)*s+(A*(M+N)+B*N)*s^2+R*(M*A*B+(M+N)*A*C+N*B*C)*s^3+(M*N*A*B)*s^4+(A*B*C*M*N)*s^5);

poles(H,s)

ans =

- 0.039821880406990304571458835810199 + 4.4322993309209319898785104967336i

- 0.039821880406990304571458835810225 - 4.4322993309209319898785104967336i

-0.074481183590774866175653557958632

- 2.4229375277976222623407143852105 - 39.657656365577651242370854262579i

- 2.4229375277976222623407143852105 + 39.657656365577651242370854262579i

And here is the code that doesn't:

>> clear;

syms R A B C M N s;

assume(in(R , 'real') & R>0);

assume(in(A , 'real') & A>0 & A<1);

assume(in(B , 'real') & B>0 & B<1);

assume(in(C , 'real') & C>0 & C<1);

assume(in(M , 'real') & M>0 & M<1);

assume(in(N , 'real') & N>0 & N<1);

H = R/(1+R*(A+B+C)*s+(A*(M+N)+B*N)*s^2+R*(M*A*B+(M+N)*A*C+N*B*C)*s^3+(M*N*A*B)*s^4+(A*B*C*M*N)*s^5);

poles(H,s)

Warning: Unable to determine poles.

> In sym/poles (line 101)

ans =

Empty sym: 0-by-1

Is there a way to make the second code work?

Thanks!

[David Goodmanson]

Hi Sita,

In this case solving for the poles is the same as solving (algebraically) for the zeros of a fifth degree polynomial in x, with arbitrary coefficients. For a polynomial of degree > 4, this is not possible.

[Sita]

Thanks everyone for your answers. I was thinking it might be impossible, so it's good to know that that's the case. I can settle for plugging in values and having it calculate numerical poles to find limits.

Answer 1

[John D'Errico]

You need to understand that this becomes a problem of solving for the roots of a general 5th degree polynomial. If ALL of the coefficients of that polynomial are known, as NUMBERs, then the solver can find the roots using numerical methods. But if even ONE of the variables is an unknown, then it is provable in general that no solution can be determined, except for some trivial cases. Sorry, but this result has been known for well over a two hundred years.

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

So no, it is absolutely impossible to solve the second problem as posed. This is not a problem in MATLAB. Some other tool, perhaps Mathematica, will still be unable to solve a problem that is mathematically impossible to solve.