solving large number of simultaneous linear equations

Suppose I am solving the following problem:

dK=sym('dK',[2 2]);

I_LHS=[3 4;2 2]*dK*[1 2;1 1];
I_RHS=[36 47;20 26];

eqn1=I_LHS(1,1)==I_RHS(1,1);eqn2=I_LHS(1,2)==I_RHS(1,2);
eqn3=I_LHS(2,1)==I_RHS(2,1);eqn4=I_LHS(2,2)==I_RHS(2,2);

eqns=[eqn1,eqn2,eqn3,eqn4];

Sol=vpasolve(eqns);
dK=double([Sol.dK1_1,Sol.dK1_2;Sol.dK2_1,Sol.dK2_2])

the solution it is giving is dK=[1 3;2 4], which is absolutely fine. Now for my actual larger problem (having 11200 equations and 11200 variables) it is really difficult to implement this way, because I have to write eqn1=I_LHS(1,1)==I_RHS(1,1);........;eqn11200=I_LHS(20,560)==I_RHS(20,560); and eqns=[eqn1,.....;eqn11200]; and so for dK of last line of code. Note that obtaining dK is just a part of my original code. In fact for this large problem MATLAB shows: " earlier syntax errors confuse code analyzer (or a possible analyzer bug)" at the end of for loop of the code. Can anyone help me to solve this problem is a better way?

Answer 1

[Walter Roberson]

 eqns = I_LHS - I_RHS;
 sol = solve(eqns);
 dK = reshape( structfun(@double, sol), size(I_LHS.')) .';

Notice the two transposes. My tests indicate that the field names of the sol structure would be in the order

     dK1_1: [1×1 sym]
     dK1_2: [1×1 sym]
     dK2_1: [1×1 sym]
     dK2_2: [1×1 sym]

which is row order fastest. To convert to column order fastest as is needed for your dK array, you reshape with the number of columns as the first size, and then transpose.

[Kamal Bera]

Thanks for the quick responses. This will solve my problem.

[Bjorn Gustavsson]

Why such an obfuscated solution? Why not using something that looks sensible:

M1 = vpa([3,4;2 2]);
M2 = vpa([1 2;1 1]);
I_RHS = vpa([36 47;20 26]);
dK = M1\(I_RHS/M2);

Nice neat clean, linear equations solved as linear equations should be solved.

[Walter Roberson]

In my opinion, using both mldivide and mrdivide in an expression is more obfuscated than solve.

Answer 2

[Bjorn Gustavsson]

If you simply use the standard numerical matrix capabilities of matlab something like this should be your solution:

% I_LHS == [3 4;2 2]*dK*[1 2;1 1]
M1 = [3 4;2 2];
M2 = [1 2;1 1];
% I_LHS == M1*dK*M2
% I_LHS/M2 == M1*dK
% M1\(I_LHS/M2) == dK
dK = M1\(I_LHS/M2)

Provided that your larger matrices are well-conditioned.

HTH