‘Polynomial expression expected.’ showed when i tried to get the coefficents

syms T p1 p2 p3 p4 p5 p6 real
J = T + T*((6*p1)/T^2 - (2*p4)/T)^2 + T*((6*p2)/T^2 - (2*p5)/T)^2 + T*((6*p3)/T^2 - (2*p6)/T)^2 + (T^6*((12*p1)/T^3 - (6*p4)/T^2)^2)/3 + (T^6*((12*p2)/T^3 - (6*p5)/T^2)^2)/3 + (T^6*((12*p3)/T^3 - (6*p6)/T^2)^2)/3 - T^2*((6*p1)/T^2 - (2*p4)/T)*((12*p1)/T^3 - (6*p4)/T^2) - T^2*((6*p2)/T^2 - (2*p5)/T)*((12*p2)/T^3 - (6*p5)/T^2) - T^2*((6*p3)/T^2 - (2*p6)/T)*((12*p3)/T^3 - (6*p6)/T^2);
J1 = diff(J,T);
cT = coeffs(J1,T);

Just got J for another window.

why would it show "Error using symengine Polynomial expression expected." as i ran it?

should i get a normal form of J1 first, when i want the coefficients of it? If so, how?

Appreciate for an answer.

Answer 1

[Walter Roberson]

coeffs() can only be used for polynomials with respect to the variable. Your expression has division by your variable, and so is not a polynomial.

simplify() would bring a normal form, but it would still have division by the variable and so is not enough.

What you can do is

[N, D] = numden(J1);
Nc = coeffs(N, T, 'all');
Dc = coeffs(D, T, 'all');

Now Nc are the numerator coefficients and Dc are the denominator coefficients.

[Zhanhao Liang]

Thx for that.

so that means, here in Matlab, there is no way to get a normal form of a polynomial,whose expression contains divided variables, right?

[Walter Roberson]

By definition, polynomials never have division by the variable that they are polynomials in, so the situation can never occur.

Your J1 can be expressed as a ratio of polynomials, but not as a polynomial. simplify() will (in this case) bring you the ratio, but you cannot directly coeffs() it because coeffs() is only for polynomials and you do not have a polynomial. I show above how to extract the two polynomials and take their coefficients individually.

[Zhanhao Liang]

Appreciate that, Mr Roberson.

I have noticed that Nc is exactly the coefficients of th polynomial that simplify(J1) presents.

>> Nc
 
Nc =
 
[ 24*p4^2 + 24*p5^2 + 24*p6^2, 1 - 48*p2*p5 - 48*p3*p6 - 48*p1*p4, 0, 8*p4^2 + 8*p5^2 + 8*p6^2, - 72*p1*p4 - 72*p2*p5 - 72*p3*p6, 108*p1^2 + 108*p2^2 + 108*p3^2]

>> simplify(J1)
 
ans =
 
(24*T^5*p4^2 + 24*T^5*p5^2 + 24*T^5*p6^2 - 48*T^4*p1*p4 - 48*T^4*p2*p5 - 48*T^4*p3*p6 + T^4 + 8*T^2*p4^2 + 8*T^2*p5^2 + 8*T^2*p6^2 - 72*T*p1*p4 - 72*T*p2*p5 - 72*T*p3*p6 + 108*p1^2 + 108*p2^2 + 108*p3^2)/T^4

If i am right, mathematically simplify(J1) == 0 would be :

Nc(1)*T+Nc(2)*1+Nc(3)*T^(-1)+...+Nc(6)*T^(-4) = 0

What i want to find is the analytical expression of T in J1, so it seems ok to use roots(NC) to get that, given that roots() is to solve a equation like this:

Nc(1)*T^5+Nc(2)*T^4+Nc(3)*T^3+...+Nc(6)*1 = 0

That's exactly what i want to do.

However, z1 in the result of roots(Nc) confuses me:

>> roots(Nc)
 
ans =
 
 root(z1^5 - (z1^4*(48*p1*p4 + 48*p2*p5 + 48*p3*p6 - 1))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (z1^2*(8*p4^2 + 8*p5^2 + 8*p6^2))/(24*p4^2 + 24*p5^2 + 24*p6^2) - (z1*(72*p1*p4 + 72*p2*p5 + 72*p3*p6))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (108*p1^2 + 108*p2^2 + 108*p3^2)/(24*p4^2 + 24*p5^2 + 24*p6^2), z1, 1)
 root(z1^5 - (z1^4*(48*p1*p4 + 48*p2*p5 + 48*p3*p6 - 1))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (z1^2*(8*p4^2 + 8*p5^2 + 8*p6^2))/(24*p4^2 + 24*p5^2 + 24*p6^2) - (z1*(72*p1*p4 + 72*p2*p5 + 72*p3*p6))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (108*p1^2 + 108*p2^2 + 108*p3^2)/(24*p4^2 + 24*p5^2 + 24*p6^2), z1, 2)
 root(z1^5 - (z1^4*(48*p1*p4 + 48*p2*p5 + 48*p3*p6 - 1))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (z1^2*(8*p4^2 + 8*p5^2 + 8*p6^2))/(24*p4^2 + 24*p5^2 + 24*p6^2) - (z1*(72*p1*p4 + 72*p2*p5 + 72*p3*p6))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (108*p1^2 + 108*p2^2 + 108*p3^2)/(24*p4^2 + 24*p5^2 + 24*p6^2), z1, 3)
 root(z1^5 - (z1^4*(48*p1*p4 + 48*p2*p5 + 48*p3*p6 - 1))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (z1^2*(8*p4^2 + 8*p5^2 + 8*p6^2))/(24*p4^2 + 24*p5^2 + 24*p6^2) - (z1*(72*p1*p4 + 72*p2*p5 + 72*p3*p6))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (108*p1^2 + 108*p2^2 + 108*p3^2)/(24*p4^2 + 24*p5^2 + 24*p6^2), z1, 4)
 root(z1^5 - (z1^4*(48*p1*p4 + 48*p2*p5 + 48*p3*p6 - 1))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (z1^2*(8*p4^2 + 8*p5^2 + 8*p6^2))/(24*p4^2 + 24*p5^2 + 24*p6^2) - (z1*(72*p1*p4 + 72*p2*p5 + 72*p3*p6))/(24*p4^2 + 24*p5^2 + 24*p6^2) + (108*p1^2 + 108*p2^2 + 108*p3^2)/(24*p4^2 + 24*p5^2 + 24*p6^2), z1, 5)
 

I have read a coupl of answers that you have done for similar questions about root with z1, but still can't get it.

Here https://www.mathworks.com/matlabcentral/answers/49689-solving-systems-of-equuations-receiving-z1-in-solution, you have mentioned that if z appears with 5 order, then the root could not be found analytically, is that exactlly the situation in my case here?

If so, how can i get a "readable" form of the results from roots(Nc) above? Numerical form in parameters p1,...,p6 will be ok.

Thanks a lot,

Zhan

[Walter Roberson]

"I have noticed that Nc is exactly the coefficients of th polynomial that simplify(J1) presents."

No. J1 is not a polynomial. J1 goes to infinity at T=0 provided that the coefficients are real valued, but polynomials are always finite valued for finite values of their variables (unless the coefficients become infinity somehow)

What you noticed was something about the numerator of the expression.

[Walter Roberson]

you are correct, there is no analytic solution for J1 = 0, and so no readable form of the solutions. For any given numeric p* values, numeric solutions can be obtained.

[Zhanhao Liang]

Get that. Sorry about being unclear about the definition of polynomial.

But what my goal is to get a parametric analytical expression of T from J1 == 0, so i wrote this:

[solx, params, conds] = solve(J1, T, 'ReturnConditions', true);

>> solx
 
solx =
 
z1

how to check z1 then?

[Walter Roberson]

look at conds. It is unlikely to tell you anything more useful than T cannot be 0.

[Walter Roberson]

it is a mess of solutions that appear to mostly reduce down to saying that the individual 5 roots are valid provided that they are real valued.

[Zhanhao Liang]

Thx i have looked at it, but it is not very helpful.

So, i know that J1 is not a polynomial, but given that the dominator of variable of J1 is in form of T^n, we can trasform the left hand side of J1 == 0 into polynomial through multiplying the highest order of T^n.

Can it not be done here in matlab?

[Walter Roberson]

Yes, and the answer you get would N (the first output of the numden call).

You can remove a lot of the distraction on the solve() by not declaring your p symbols to be real valued. When you solve() you then do not get any warning about needing to return conditions, and you get

 root(24*p6^2*z^5 + 24*p5^2*z^5 + 24*p4^2*z^5 - 48*p3*p6*z^4 - 48*p2*p5*z^4 - 48*p1*p4*z^4 + z^4 + 8*p6^2*z^2 + 8*p5^2*z^2 + 8*p4^2*z^2 - 72*p3*p6*z - 72*p2*p5*z - 72*p1*p4*z + 108*p3^2 + 108*p2^2 + 108*p1^2, z, 1)

repeated up to root number 5. And there is not obvious way to reduce that to lower degree.

The problem is not in isolating something that looks polynomial-like: the problem is that your J is too high a degree to be able to find a closed form solution for its critical points in terms of T.

[Zhanhao Liang]

Very appreciate for your patience and specific answer, Mr Roberson.

I have decided to use numerical method in my coding part to give me a bridge.

We tallk next time!