I have an application in which I am computing a quantity which is approximated by an average over $M$ points. In theory, the average converges to the correct quantity when $M$ is infinite. In practice, the computation involves a $M\times M$ symmetric positive definite matrix, whose condition number gets huge when $M$ increases. I compute the condition number as the ratio of the biggest and smallest eigenvalues given by a selfadjoint eigensolver.
Therefore, the average I'm computing does not seem to reach stationarity when I increase $M$, but instead starts deviating quite a bit from what I feel is the correct value when $M$ is really big.
How can I choose the best $M$? Is there a way to put some error bars on the result I get, based on the condition number? What is a bad condition number for the $LDL^T$ cholesky decomposition and for a SPD matrix, e.g. when should I start worrying?