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

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Bill Tubbs
Bill Tubbs 2021-3-5
编辑： Paul 2022-7-18
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

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Paul
Paul 2022-7-14
Ran in:
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
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Bruno Luong
Bruno Luong 2022-7-17
Ran in:
@Paul But, I am surprised by this ...
It looks like the discrepency occurs when matrix size is between 2 and 9, beyond that pagemldivide seems to match backslash.
nmax = 100;
errx = zeros(1,nmax);
for n=1:nmax
A = rand(n,n,10);
B = rand(n,n,10);
X = pagemldivide(A,B);
errx(n) = norm(X(:,:,end)-A(:,:,end)\B(:,:,end));
end
plot(1:nmax, errx);
xlabel('Matrix size');
ylabel('Error betwen pagemldivide');
Paul
Paul 2022-7-18
编辑：Paul 2022-7-18
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.

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