# Solving for coefficients in polynomial

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Benjamin on 3 Apr 2019
Commented: Matt J on 4 Apr 2019
I have the following equation in MATLAB which solves for my coefficients:
A45 = [(eta./eta_c).^(4:7).*(4:7)]\(Z_MD - Zfixed); % Solve for the higher order coefficients. This is the norm solution.
This represents the summation part of this equation: I set the first 3 coefficients which is why it just goes from 4 to 7.
As you can see, since the equation is a sum over k*A_k*(eta./eta_c)^k. I believe the above equation solves for each A_k, correct? i think it does, but I'm not sure how it does it. The A_k's are not even in the MATLAB equation. How does it know to create these coeffs and solve for them?
Also, what if I wanted to have order 4-7 polynomial, but then I wanted to skip 8 and 9 and do 10? How would I do that?
The following does not work:
A45 = [(eta./eta_c).^(4:7,10).*(4:7,10)]\(Z_MD - Zfixed); % Solve for the higher order coefficients. This is the norm solution.

Matt J on 4 Apr 2019
Edited: Matt J on 4 Apr 2019
How does it know to create these coeffs and solve for them?
In Matlab, if you have a matrix equation M*x=z, you can solve for x by doing
x=M\z;
In the special case where M is a square non-singular matrix, this is similar to doing x=inv(M)*z, but better. This is all that the code you've posted is doing, for a particular choice of the matrix M and z.
How does it know how many x(i) to solve for? From the number of columns of the matrix M.

Benjamin on 4 Apr 2019
Ok cool, so just to be clear. If I have:
A45 = [(eta./eta_c).^([4:7,10,22]).*([4:7,10,22])]\(Z_MD - Zfixed);
Then this would sove for A4, A5, A6, A7,A10, and A22 correct? I am just ignoring the other terms (A8, A9, A11-A21).
Matt J on 4 Apr 2019
if you are assuming that A8, A9, A11-A21 are all zero, then yes, what you say is correct.

Walter Roberson on 3 Apr 2019
k = 4:7;
A45 = [k .* A(k) .* (eta./eta_c).^k]\(Z_MD - Zfixed); % Solve for the higher order coefficients. This is the norm solution.
and you could also use k = [4:7, 10]

Matt J on 3 Apr 2019
I think Walter really meant to write,
Ak = [k .* (eta./eta_c).^k]\(Z_MD - Zfixed);
Walter Roberson on 4 Apr 2019
No. I have no idea why that \ is being done; the original equation has no obvious polynomial. But the original equation does have a sum of A_k times those factors, so A(k) should appear in the list.
Benjamin Cowen on 4 Apr 2019
It is a polynomial. Eta is the variable being squared, cubed, etc. I’m only caring about the summation term not the others, so it is a polynomial. Eta_c is a constant