MATLAB: Outputting all possible solutions to system of equations with constraints.

cell arraysequationmatrixmultiplenonlinearsolvesymbolictable

I'm trying to solve a system of nonlinear equations for which I have more unknowns than equations. I have conditions for a few of the variables to limit the number of solutions. And I would also like to output the solutions into an easily visually parseable table of numeric values where all values are aligned to their respective solutions.
What I have to begin with is the code below. This is giving me a parameterized solution warning, though. Assuming I haven't made a syntax mistake this should produce a solution. I have tested a solution by explicitly declaring variables within the bounds I've set to confirm this. For example, if:
V_as = 275
f_s = 140
then without conditions MATLAB would return:
C_ms = 836e-6
BL = 3.41
Q_ms = 3.8
M_md = 1.48e-3
Maybe just setting these boundaries still isn't constraining it enough? Should I try specifically declaring an array of values for f_s and V_as? Would that help? And how would I go about doing that?
Also, I would like to output all these variables into a table of numeric values aligned to each solution but I'm not sure how to do that since they're stored in a symbolic cell array. Something like:
 output_table = [double(sol.BL), double(sol.M_md), double(Q_ms), etc...]
And how to make sure that each variable in a particular row is from the same solution as the variable next to it in the same row? And how to set the columns to have headers representing each variable?
syms f_s V_as M_ms Q_es Q_ms C_ms M_md BL;
D = 4.4e-2;
S_d = pi*(D/2)^2;
SPL = 85;
Q_ts = 0.35;
R_e = 3.3;
assume(in(f_s/10,'integer') & f_s>=100 & f_s<=150);
assume(in(V_as/25,'integer') & V_as>=100 & V_as<=1000);
assume(M_md>=1e-3 & M_md<5e-3);
assume(BL>3 & BL<3.5);
assume(C_ms>0 & C_ms<1500e-6);
assume(Q_ms>0 & Q_ms<10);
e1 = f_s == 1/(2*pi*sqrt(M_ms*C_ms));
e2 = Q_es == R_e/BL^2*sqrt(M_ms/C_ms);
e3 = Q_ts == Q_ms*Q_es/(Q_ms+Q_es);
e4 = V_as*10^-6 == 1.21*343^2*S_d^2*C_ms;
e5 = M_ms == S_d^2*(M_md/S_d^2+2*8*1.21/(3*pi^2*(D/2)));
e6 = SPL == 20*log10(1.21*BL*S_d*sqrt(R_e)/(2*pi*R_e*M_ms*2e-5));
sol = solve(e1,e2,e3,e4,e5,e6,f_s,V_as,Q_es,Q_ms,M_ms,M_md,BL,C_ms);

Best Answer

There do appear to be algebraic solutions, but they are numerically unjustifiable for your inputs. Your data such as D = 4.4e-2 has between 1 and 4 significant figures, so you cannot justify calculations that carry more than at most 4 significant figures. However, to achieve the conditions that in(f_s/10,'integer') and in(V_as/25,'integer') then because of the Pi and the various squares and sqrt(), some of your values such as C_ms and BL must be infinitely precise, such as BL must be exactly (14235529/62500)*sqrt(330^(1/17)*sqrt(10))*330^(8/17)/(f_s_10^2*V_as_25) where f_s_10 and V_as_25 are integers
One of the algebraic solutions appears to be at f_s = 150, V_as = 250; there are other algebraic solutions as well. But they require infinite precision for the integer constraints to be met, and they require assuming that the various low-precision constants such as D = 4.4e-2 are "actually" infinitely precise such as D = 44/1000 and not D = (44/1000 +/- 5/10000) as should be implied by the significant figures.
Related Question