In my numerical linear algebra class, I learned that for some matrices, it could have an element that is a very small number that is approximately 0 (and many orders of magnitude different from all the other non-zero elements), can destroy the conditioning of a matrix.
Now I am wondering how this works practically. When we formulate the linear system $$Ax=b$$
A lot of math goes into forming the operator $A$. In the case of solving PDEs, we run into a lot of round-off issues during computations, and this, will certainly affect $A$. In some of the matrices I've formed, my operator is a sparse matrix. I notice that some elements of $A$ end up being approximately 0, but not actual zero, decreasing the sparsity, and perhaps destroying the conditioning of the matrix? Theoretically, those computations should have evaluated zero, but due to roundoff, they end up non-zero.
Would it be a good approach to go through the non-zero elements of a sparse matrix and filter out all the values close to zero and set them as zero? I have thought about doing this, but I am unsure if this is the right approach. What if those numbers that I am filtering are not theoretically zero, and they are actually very small?