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function x = kronm(Q,x)
% Fast Kronecker matrix multiplication, for both full and sparse matrices
% of any size. Never computes the actual Kronecker matrix and omits
% multiplication by identity matrices.
% y = kronm(Q,x) computes
% y = (Q{k} kron ... Q{2} kron Q{1})*x
% If Q contains only two matrices and x is a vector, the code uses the
% identity
% ( Q{2} kron Q{1} )*vec(X) = vec(Q{1}*X*Q{2}'),
% where vec(X)=x. If Q contains more than two matrices and/or if x has more
% than one column, the algorithm uses a generalized form of this identity.
% The idea of the algorithm is to see x as a multi-dimensional array and to
% apply the linear maps Q{i} separately for each dimension i. If Q contains
% just one matrix, the function returns the regular matrix product Q{1}*x.
%
% Inputs:
% Q: 1-by-k cell array containing k matrices of arbitrary size
% (can be sparse). Denote by R(i) the number of rows of Q{i}, and
% by C(i) the number of columns. Alternatively, Q{i} may also be
% a scalar qi. This is interpreted as the qi-by-qi identity
% matrix. Hand over identity matrices in this fashion for optimal
% performance.
% x: Matrix of size CC-by-m, where CC=C(1)*...*C(k).
%
% Output: Matrix of size RR-by-m, where RR=R(1)*...*R(k).
%
%
% Example:
% R = [60, 30, 20]; % Number of rows for matrices Q{1},Q{2},Q{3}.
% C = [55, 25, 15]; % Number of columns of matrices Q{i}.
% m = 5; % Number of columns of x.
% Q = cell(1,length(R)); % Create cell with sparse random matrices
% for i=1:length(R) % of density 0.05.
% Q{i} = sprand(R(i),C(i),0.05);
% end
% x = rand(prod(C),m); % Random matrix x with C(1)*C(2)*C(3) rows.
% y = kron(Q{3},kron(Q{2},Q{1}))*x;
% % Matlab's Kronecker multiplication...
% yy= kronm(Q,x); % and kronm...
% norm(y-yy) % ... give the same result up to
% % computational error.
%
%
% Version: 6-Oct-2015
% Author: Matthias Kredler (Universidad Carlos III de Madrid)
% mkredler@eco.uc3m.es
% Acknowledgement:
% This code follows the same idea as 'kronmult' by Paul G. Constantine &
% David F. Gleich (Stanford, 2009). However, I avoid loops and allow for
% non-square inputs Q{i}. I have also included the special treatment for
% identity matrices.
m = size(x,2); % Obtain number of columns in input.
k = length(Q); % Number of matrices in Q.
R = zeros(1,k); % Vector for number of rows of,
C = zeros(1,k); % Q-matrices and for number of columns.
comp = true(1,k); % Check if we have to multiply by Q{i}.
for i=1:k
if isscalar(Q{i}) % If input Q{i} is a scalar, don't
comp(i) = false; % have to multiply in this dimension.
R(i) = Q{i}; % Read in number of rows and columns.
C(i) = Q{i};
else % Otherwise, read out size of the
[R(i),C(i)] = size(Q{i}); % matrix.
end
end
xsiz = [C,m]; % Will constantly change dimension of x.
% xsiz is the current size, when x is
% reshaped to array of dim.
% C(1),C(2),...,C(k),m.
if comp(1) % Start with first Kronecker product,
x = Q{1}*reshape(x,[C(1),prod(xsiz)/C(1)]);
% leave out if Q{i} is identity.
xsiz(1) = R(1); % Replace size of dimension 1.
end % (Don't do this in loop below --> save
% time on reshapes and permutes)
if k>1 && m==1 % If Q has just one element, we're done.
if comp(k) % If x was a column vector, do the last
x = reshape(x,[prod(xsiz)/C(k),C(k)]) *Q{k}' ;
xsiz(k) = R(k); % Kronecker product by matrix
end % post-multiplication to save time on
% reshapes and permutes.
loopTo = k-1; % Will only have to loop up to
% dimension k-1 below.
else % If x is a matrix, have to loop over
loopTo = k; % all dimensions.
end
if k>2 || m>1 % Now loop over remaining dimensions,
x = reshape(x,xsiz); % inf any. Reshape x into an array of
for i=2:loopTo % dimension R(1),C(2),...,C(k)or R(k),m.
if comp(i) % If Q{i} is not identity: Create
dims = 1:k+1; % vector to re-shuffle dimensions.
dims(i) = []; % Put dimension i first (by permute),
dims = [i, dims]; %#ok<AGROW> % e.g. order [2,1,3,4,5]
% for i=2 and k=4. Turn off Matlab's
% warning for size change.
Xmat = reshape( permute(x,dims), [C(i), prod(xsiz)/C(i)] );
% Then bring array into matrix with
Xmat = Q{i}*Xmat; % N(i) rows, ex: N(2)-by-N(1)*N(3)*...
% *N(4)*m and multiply by Q{i}.
xsiz(i) = R(i); % Changed dimensionality of x.
x = ipermute( reshape(Xmat,[R(i), xsiz(dims(2:k+1))]), dims );
end % Reshape back to array, ex: to dim.
end % N(2),N(1),N(3),N(4),m, and inverse-
% permute to go back to orginal array,
end % ex: dim. N(1),N(2),N(3),N(4),m.
x = reshape(x,[prod(R),m]); % Then give back result as matrix.
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