Consider a tridiagonal matrix with diagonal entries equal to 10 and subdiagonal and superdiagonal entries -3.5. I'm working on an iterative method where I can solve $Ax=b$ using the iteration $x_{i+1} = x_i + \Xi(b-Ax_i),$ where $\Xi \in \mathbb{R}.$
I want to find which values of $\Xi$ there are such that we converge to the solution $\tilde{x}$ such that $A\tilde{x} = b.$ I'm wondering if I might be able to figure out the $\Xi$'s by pulling some of the ideas from the standard Thomas algorithm, but otherwise I am having difficulty figuring out how the properties of a tridiagonal matrix will infer any information about $\Xi$.
I was wondering if I could get some suggestions on this problem.