Best way to calculate the fundamental matrix of an absorbing Markov Chain?

Question

I have a very large absorbing Markov chain (scales to problem size -- from 10 states to millions) that is very sparse (most states can react to only 4 or 5 other states).

<

Answer 1

The reason you're getting the advice not to use matrix inverses for solving equations is because of numerical stability. When you're matrix has eigenvalues that are zero or near zero, you have problems either from lack of an inverse (if zero) or numerical stability (if near zero). The way to approach the problem, then, is to use an algorithm that doesn't require that an inverse exist. The solution is to use Gaussian elimination. This doesn't provide a full inverse, but rather gets you to row-echelon form, a generalization of upper-triangular form. If the matrix is invertible, then the last row of the result matrix contains a row of the inverse. So just arrange that the last row you eliminate on is the row you want.

I'll leave it to you to understand why I-Q is always invertible.