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

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

Answer 1

[Paul]

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

[Bill Tubbs]

As you suggest, the result is more complex than the for loop and probably not much more efficient. I recently upgraded to 2021b so I'll have to wait but this is good to know about. Thanks.

[Bill Tubbs]

So they introduced a new set of functions to do matrix algebra along the 3rd dimension of arrays (pagemrdivide | pagemldivide | pagemtimes | pagetranspose etc)?

Any idea why they didn't use array broadcasting as in Python and Julia? In Python it broadcasts all the arrays into the higher dimension automatically allowing you to use the existing linear algebra operators and functions.

Unvectorized version in Python (similar to MATLAB for loop above):

V = np.hstack([np.linalg.solve(Aa - L[i] * Ia, -Ba * Phi[i])
               for i in range(nv)])

Vectorized version:

V = np.linalg.solve(
    Aa - L.reshape(-1, 1, 1) * Ia, 
    -Ba * Phi.reshape(-1, 1, 1)
).reshape(2, 2).T

[Note the L.reshape(-1, 1, 1) rather than reshape(L,1,1,[]) as in MATLAB. In Python the two highest dimensions are assumed to be the ones where matrix operations occur not the first two.]

[Paul]

pagemldivide does support implicit expansion. It also works for N-D arrays and supports transpose operations on the inputs. Maybe that's why new functions were developed instead of using the existing mldivide, etc.

rng(102)
r = rand(2,2,2);
R = rand(2);
ri = pagemldivide(r,R);
format long e

But, I am surprised by this

ri(:,:,1) - r(:,:,1)\R

ans = 2×2

1.0e+00 * 0 -5.551115123125783e-17 2.775557561562891e-17 1.110223024625157e-16

ri(:,:,2) - r(:,:,2)\R

ans = 2×2

1.0e+00 * 1.110223024625157e-16 -9.714451465470120e-17 -3.330669073875470e-16 4.440892098500626e-16

[Bruno Luong]

@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]

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.