Block matrix inner products in Matlab

0
votes

I have been using the following custom function to perform the multiplication of a vector by a matrix, in which each element of the vector multiplies a 3x3 block within a (3xN)x(3) matrix:

function [B] = BlockScalar(v,A)

 N=size(v,2);
 B=zeros(3*N,3);

 for i=1:N
     B(3*i-2:3*i,:) = v(i).*A(3*i-2:3*i,:);
 end

end

Similarly, when I want to multiply a collection of 3x3 matrices by a collection of 3x3 vectors, I use the following

function [B] = BlockMatrix(A,u)

 N=size(u,2);
 B=zeros(N,3);

 for i=1:N
     B(i,:) = A(3*i-2:3*i,:)*u(:,i);
 end

end

Since I call them very often, these unfortunately, slow down the running of my code significantly. I was wondering if there was a more efficient (perhaps vectorised) version of the above operations.

2
matlabmatrixlinear-algebra

2 Answers

2
votes

In both instance, you are able to do away with the for-loops (although without testing, I cannot confirm if this will necessarily speed up your computation).

For the first function, you can do it as follows:

function [B] = BlockScalar(v,A)
% We create a vector N = [1,1,1,2,2,2,3,3,3,...,N,N,N]
N=ceil((1:size(A,1))/3); 

% Use N to index v, and let matlab do the expansion
B = v(N).*A;

end

For the second function, we can make a block-diagonal matrix.

function [B] = BlockMatrix(A,u)

 N=size(u,2)*3;
 % We use a little meshgrid+sparse magic to convert A to a block matrix
 [X,Y] = meshgrid(1:N,1:3);

% Use sparse matrices to speed up multiplication and save space
 B = reshape(sparse(Y+(ceil((1:N)/3)-1)*3,X,A) * (u(:)),3,size(u,2))';

end

Note that if you are able to access the individual 3x3 matrices, you can potentially make this faster/simpler by using the native blkdiag:

function [B] = BlockMatrix(a,b,c,d,...,u)
% Where A = [a;b;c;d;...];

% We make one of the input matrices sparse to make the whole block matrix sparse
% This saves memory and potentially speeds up multiplication by a lot
% For small enough values of N, however, using sparse may slow things down.
reshape(blkdiag(sparse(a),b,c,d,...) * (u(:)),3,size(u,2))';

end
0
votes

Here are vectorized solutions:

function [B] = BlockScalar(v,A)
    N = size(v,2);
    B = reshape(reshape(A,3,N,3) .* v, 3*N, 3);
end


function [B] = BlockMatrix(A,u)
    N = size(u,2);
    A_r = reshape(A,3,N,3);
    B = (A_r(:,:,1) .* u(1,:) + A_r(:,:,2) .* u(2,:) + A_r(:,:,3) .* u(3,:)).';
end

function [B] = BlockMatrix(A,u)
    N = size(u,2);
    B = sum(reshape(A,3,N,3) .* permute(u, [3 2 1]) ,3).';
end