how to fprint a particular element corresponding to other element

I need to solve for a M value, so I created an array of M values, one of which is the correct M. The M is imbedded multiple times in a function. I know my function has to equal 1.2. Once I ran the code I had had many solutions in an matrix, I was able to find an answer close to 1.2. My question is, is there a way I can directly fprint what M I used to acquire the correct answer.

%%Constants
r   = 1.4;
rm1 = r-1;
rp1 = r+1;

%%Process
M=[1:.01:3]
A_AS=sqrt((1./M.^2).*(((2+rm1.*M.^2)/rp1).^(rp1/rm1)));
Po2_Po1=((rp1*.5*M.^2)./(1+(rm1*.5*M.^2))).^(r/rm1).*(1./(((2*r)/rp1).*M.^2-(rm1/rp1))).^(1/rm1)
Ac_At=A_AS.*Po2_Po1

I would appreciate the help, thank you for your time.

Answer 1

[Walter Roberson]

[~, idx] = min(abs(Ac_at - 1.2));
M(idx)

By the way, exact solution is

temp = sqrt(roots([15593, -72030, 56595, -16660, 2415, -174, 5]));
temp(imag(temp)~=0) = [];
temp(temp < 1 | temp > 3) = [];
M = temp

[Jose Camorlinga Medina]

Very helpful, thank you.

[Jose Camorlinga Medina]

what do the numbers inside the root represent?

[Walter Roberson]

For r = 1.4 only, you are trying to solve

216*M^6*(1/(7*M^2-1))^(1/2)/((M^2+5)^(1/2)*(7*M^2-1)^2) == Value

where Value == 1.2 in your case.

This has a solution form of

RootOf((16807*Value^2-46656) * z^12 + 72030*Value^2 * z^10 -56595*Value^2 * z^8 + 16660*Value^2 * z^6 -2415*Value^2 * z^4 + 174*Value^2 * z^2 -5*Value^2, z)

where RootOf(f(z),z) stands for the set of values, z, such that f(z) is 0 -- the roots of the polynomial.

You can see that only even powers are used, so this can be reduced to +/- the square root of the roots of a degree 6 polynomial,

+/- sqrt( roots([(16807*Value^2-46656), 72030*Value^2, -56595*Value^2, 16660*Value^2, -2415*Value^2, 174*Value^2, -5*Value^2]) )

Again, Value here is the 1.2 that you are seeking.