Question

There is a large matrix that is used in Ax=b type problem. A is symmetric. Is there any algorithm to let us save only half of the matrix and do operation like x=A\b on it?

Answer 1

You’ll only save half the memory, but you can do this by creating a flat version of the matrix, saving that, then unflattening it. The extra time required probably doesn’t make the saving worthwhile, mind:

% pretend this is symettric...
A = rand(10, 10);

% store it as a flat list
flatA = [];
for k = 1:size(A,1)
    flatA = [flatA; A([1:k],k)];
end

save('flatA.mat', 'flatA');

% undo
N = (-1 + (1+4*2*length(flatA))^0.5)/2;
newA = NaN(N,N); 
cur = 1;
for k = 1:N
    len = k;
    newA(1:len, k) = flatA(cur:cur+len-1);
    cur = cur + len;
end
% real A cannot have NaNs or this trick fails
newA(isnan(newA)) = newA(isnan(newA'))';