# Is it possible to solve multiple linear systems of equations in parallel with one matrix operation?

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Bill Tubbs on 5 Mar 2021
Edited: Paul on 18 Jul 2022
I'm wondering if there's a way to do the following calculations in one go (i.e. without the for loop).
V = nan([na nv]);
for i=1:nv
% Solve homogenous system of equations
V(:,i) = [Aa-L(i)*Ia] \ (-Ba*Phi(i));
end
For the purposes of this example, let's consider the variables having the following values:
Aa = [ 0.819 0;
-0.544 1];
na = size(Aa,2);
Ba = [1; 0];
Phi = [1 1];
L = [0.7515 0.7165];
nv = numel(L);
Ia = eye(na);
Which yields the solution:
V =
-14.8148 -9.7561
-32.4316 -18.7207

Paul on 14 Jul 2022
Hi Bill,
pagemldivide introduced in 2022a can do the trick. Whether or not this is really better than the loop ....
Aa = [ 0.819 0;
-0.544 1];
Ba = [1; 0];
Phi = [1 1];
L = [0.7515 0.7165];
V = reshape(pagemldivide(Aa - reshape(L,1,1,[]) .* eye(size(Aa)) , -Ba .* reshape(Phi,1,1,[]) ) , numel(Ba),numel(L))
V = 2×2
-14.8148 -9.7561 -32.4316 -18.7207
Paul on 18 Jul 2022
Edited: Paul on 18 Jul 2022
I as surprised by this too. Never would have occurred to me to test for different dimensions, I'm glad it occurred to you.
Further discussion here.