Need help with solving this matrix equation

Hello eveyone

Thank you all for being here and spendint time reading this. I have recently been stucked with this matrix equation.

The code is a formula which the "Final" parameter is a 6x6 matrix, where its parameters each contain 6 unknowns defined in syms section.

So overally I have 36 equations and 6 unknowns where 30 equation is probably not necessary and due to its complexity i cannot do it myself.

So i would appreciate if you guys show me solution of how to solve the eqations. solve or fsolve or any other methods in mind.

Thanks in advance.

%% Initials %%
syms c m11 m12 m13 m33 m44

CNB = [0.283810e2 0.217478e2 0.51749e1 0.1672e0 0.1051e0 -0.13248e1; 0.217478e2 0.285360e2 0.54735e1 0.252e-1 -0.2458e0 -0.11378e1; 0.51749e1 0.54735e1 0.5091929e3 0.1461e0 0.532e-1 -0.12607e1; 0.1672e0 0.252e-1 0.1461e0 0.4791e0 0.632e-1 -0.1902e0; 0.1051e0 -0.2458e0 0.532e-1 0.632e-1 0.2931e0 0.532e-1; -0.13248e1 -0.1378e0 -0.12607e1 -0.1902e0 0.532e-1 0.35093e1;];
EP0 = [0.26054e1 0.13450e1 0.15656e1 0.15e-2 -0.1103e0 -0.607e-1; 0.13450e1 0.27312e1 0.14808e1 -0.762e-1 0.1394e0 -0.92e-2; 0.15656e1 0.14808e1 0.29516e1 -0.1478e0 0.2659e0 -0.1484e0; 0.15e-2 -0.762e-1 -0.1478e0 0.10573e1 -0.41e-2 0.1034e0; -0.1103e0 0.1394e0 0.2659e0 -0.41e-2 0.7682e0 0.289e-1; -0.607e-1 -0.92e-2 -0.1484e0 0.1034e0 0.289e-1 0.7299e0;];
CPMD = [0.348547e2 0.26368e1 0.39404e1 0.4049e0 -0.2123e0 -0.9570e0; 0.26368e1 0.258356e2 0.49370e1 0.2765e0 -0.2727e0 -0.1206e0; 0.39404e1 0.49370e1 0.663384e2 -0.369e-1 -0.4342e0 0.1556e0; 0.4049e0 0.2765e0 -0.369e-1 0.1194965e3 -0.282e-1 -0.2629e0; -0.2123e0 -0.2727e0 -0.4342e0 -0.282e-1 0.1198436e3 0.9031e0; -0.9570e0 -0.1206e0 0.1556e0 -0.2629e0 0.9031e0 0.1142415e3;];
nu = 0.395e0;
s1111 = (5-4*nu)/(8*(1-nu)); s2222 = (5-4*nu)/(8*(1-nu)); s3333 = 0; s3311 = 0; s3322 = 0; s2323 = 0.5; s3131 = 0.5;
s2211 = (-1+4*nu)/(8*(1-nu)); s1122 = (-1+4*nu)/(8*(1-nu)); s2233 = nu/(2*(1-nu)); s1133 = nu/(2*(1-nu)); s1212 = (3-4*nu)/(4*(1-nu));
S = [s1111 s1122 s1133 0 0 0; s2211 s2222 s2233 0 0 0; s3311 s3322 s3333 0 0 0; 0 0 0 s2323 0 0; 0 0 0 0 s3131 0; 0 0 0 0 0 s1212;];
I6 = eye(6,6);

cm = 0.95-c; cp = 0.05; ci = c;
CI = [m11 m12 m13 0 0 0; m12 m11 m13 0 0 0; m13 m13 m33 0 0 0; 0 0 0 m44 0 0; 0 0 0 0 m44 0; 0 0 0 0 0 ((m11-m12)/2);];

%% Main %%

TP = I6 - S*inv(S+inv(CNB-EP0)*EP0);
eq1 = -(I6 + CI * inv(EP0) + CI * inv(EP0) ^ 2) * inv(EP0) * EP0;
eq2 = eq1 - I6 + S;
eq3 = eq2 ^ 2 + I6 - eq2;
TPI = I6 - S*((cp/(ci+cp))*inv(S+inv(CNB-EP0)*EP0)+eq3);
eq4 = (I6 - (((cp+ci)*(TPI))*inv(cm*I6))+(((cp+ci)*TPI)*inv(cm*I6))^2)*inv((cm*I6));
Final = EP0 - CPMD + ((cp*(CNB-CI)*TP)+((cp+ci)*(CI-EP0)*TPI))*eq4;

[Walter Roberson]

Are there known upper or lower bounds for the variables?

[Hamed Badjian]

initial guess only 0.02 30 30 10 400 0.5

[Walter Roberson]

I need to know the order of variables for that list ;-)

[Hamed Badjian]

yes sure. Its the same as i defined above. So the initial guesses are:

c m11 m12 m13 m33 m44 = 0.02 30 30 10 400 0.5

Please let me know anything else u need. And thank you again for the effort.

best

[Walter Roberson]

The best I have so far is at [0.493175072506575551, 2.20143552126730579, 1.82530305681069338, 1.18597579267751763, -6.0791056379802999, 0.64010469232764311] but it is not very good, and the variance is quite large.

[Hamed Badjian]

Oh i see. I have recently gotten an answer using fsolve using levenberg algorithm (since the number of my equations are more than the variables) and im getting:

0.14319424051923044 319.41030595162226 -19.966251030352474 52.94308886239448 289.5473552753959 6.238287386475258

which sadly does not satisfy the equation even close when I put them in the equation.

May i know the method you employed to reach your answer above?

thanks.

[Walter Roberson]

I am running a custom minimizer that I developed that does a lot of grid searching. The values above were based on one iteration when the bounds were all set at +/- 100. I am running again with +/-600 bounds, but it will take several hours.

[Hamed Badjian]

i see. thats nice.

thank you really for this effort. it means alot :)