Simultaneously inverting many matrices

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Jeong Ho
Jeong Ho 2015-6-18
评论： Tohru Kikawada 2021-1-31
Dear all, I have many 2-by-2 matrices (which are covariance matrices). I want to invert them all. I'm curious if there's an efficient way of doing this. I thought, maybe, you create a cell, in which each element is one of these matrices, and then use cellfun() in some way to do it. Quintessentially, my question is, is there a way of simultaneously inverting many matrices? I'd appreciate any and all comments. Thank you very much in advance!
Best, John

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Walter Roberson
Walter Roberson 2015-6-18
2 x 2 you might as well use the formula
D = A(1, 2, :) .* A(2, 1, :) - A(1, 1, :) .* A(2, 2, :);
V11 = -A(2, 2, :) ./ D;
V12 = A(1, 2, :) ./ D;
V21 = A(2, 1, :) ./ D;
V22 = -A(1, 1, :) ./ D;
invs = [V11, V12; V21, V22];

更多回答（4 个）

Tohru Kikawada
Tohru Kikawada 2019-4-28
编辑：Tohru Kikawada 2019-4-28
You can leverage Symbolic Math Toolbox to vectorize the calculation.
% Define size of matrices
M=2;
N=10000;
A=rand(M,M,N);
% Calculate the inverse matrices in a loop
invA_loop = zeros(size(A));
tic
for k = 1:N
invA_loop(:,:,k) = inv(A(:,:,k));
end
disp('Elapsed time in calculation in a loop:');
toc
% Calculate the inverse matrices in a vectorization
As = sym('a', [M,M]); % Define an MxM matrix as a symbolic variable
invAs = reshape(inv(As),[],1); % Solve inverse matrix in symbol
invAfh = matlabFunction(invAs,'Vars',As); % Convert the symbolic function to an anonymous function.
tic
invA_sym = reshape(invAfh(A(1,1,:),A(2,1,:),A(1,2,:),A(2,2,:)),M,M,N);
disp('Elapsed time in the vectorized calculation:');
toc
% Max difference between the results in the loop and the vectorization
disp('Max difference between the elements of the results:');
disp(max(abs(invA_loop(:)-invA_sym(:))))
Results:
Elapsed time in calculation in a loop:
Elapsed time is 0.093938 seconds
Elapsed time in the vectorized calculation:
Elapsed time is 0.001396 seconds
Max difference between the elements of the results:
3.0323e-09
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Alec Jacobson
Alec Jacobson 2020-11-14
%invA_sym = reshape(invAfh(A(1,1,:),A(2,1,:),A(1,2,:),A(2,2,:)),M,M,N);
Acell = reshape(num2cell(A,3),1,[]);
invA_sym = reshape(invAfh(Acell{:}),M,M,N);
so the code above works for M≠2
Tohru Kikawada
Tohru Kikawada 2021-1-31
Alec, this is great! Thanks for your extension!

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James Tursa
James Tursa 2015-6-18
You might look at this FEX submission by Bruno Luong for solving 2x2 or 3x3 systems:
http://www.mathworks.com/matlabcentral/fileexchange/27762-small-size-linear-solver
Hugo
Hugo 2015-6-18
In my experience, using cells is rather slow. Since your matrices are 2x2, then you could simple arrange them in a 3D array, with the first dimension representing the index of each matrix. Let's call this matrix M, which will be of size Nx2x2, N denoting the number of matrices you want to invert.
Now recall that the inverse of a matrix A=[A11,A12;A21,A22] can be computed as
[A22, -A12; -A21, A11] /(A11*A22-A12*A21)
You can implement this for all matrices as follows:
Minv = reshape([M(:,4),-M(:,2),-M(:,3),M(:,1)]./repmat(M(:,1).*M(:,4)-M(:,2).*M(:,3),1,4),N,2,2);
Hope this helps
Hugo
Azzi Abdelmalek
Azzi Abdelmalek 2015-6-18
Using cellfun will not do it simultaneously. The for loop can be faster. But if you have a Parallel Computing Toolbox, you can do it with parfor

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