Suppose I have two sparse matrices, $A$ and $B$, of size $N \times N$. They each have the same sparcity pattern ("footprint"). They each also have values which in theory should be identical, but aren't due to machine precision error.
A simple example of this in MATLAB syntax:
% Dimensions of square, sparse matrices N = 10; A = sparse(N,N); B = sparse(N,N); % Add a dozen nonzero entries at random locations numEntries = 12; rows = randi( N, numEntries,1 ); cols = randi( N, numEntries,1 ); % Populate the entries, one at a time for entry = 1:numEntries i = rows(entry); j = cols(entry); % Generate random numbers x = rand(1,1); y = rand(1,1); % Populate matrices % Note: (x^2-y^2) / (x-y) == (x+y)*(x-y)/(x-y) == x+y A(i,j) = (x^2-y^2) / (x-y); B(i,j) = x + y; end diffMat = A - B; disp(diffMat);
Obviously the values of $A$ and $B$ should be identical, but they aren't. The output of this script is as follows:
1.0e-15 * (1,2) -0.222044604925031 (2,2) 0.222044604925031 (10,3) 0.111022302462516 (8,4) -0.222044604925031 (3,5) 0.222044604925031 (10,5) 0.555111512312578 (3,6) 0.222044604925031
Suppose a pair of matrices like this that give wildly different results $x_A$ and $x_B$ for the matrix equations $A x_A = b$ and $B x_B = b$ (same $b$ in each case). Suppose I have confirmed that the values of $A$ and $B$ are identical to less than one part in 1e15 (that is,
abs( (A(A~=0)-B(B~=0)) ./ A(A~=0) ) < 1e-15). Further suppose that the estimated condition number (
condest()) of each matrix is identical.
My question: Under what circumstances could this occur?
I originally asked this question because I had this problem in my code. It turned out that my $b$ vectors were not identical due to a small typo in the code that generated them (thanks @JesseChan!).
I am still interested if and how this can occur. My guess is that very high condition numbers can produce these kinds of results, but I'm curious if anyone can produce and explain a minimum working example of this behavior.