I have been trying to find a matrix $M$ with a size 3 x 3 for an iterative method $x_k = Mx_k +b$ with a spectral radius of M greater than 1, where in two instances the iteration converges based on the $x_0$ and $b$. And also diverges based on the $x_0$ and $b$. However, I don't really know how to find one that converges, since all the theorems I found state that the spectral radius of M is less than 1, then the iterative method converges. Is there any examples you guys can give me?
2026-03-26 22:52:30.1774565550
Find a matrix M for an iterative method with spectral radius greater than 1
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Here is an example: take $$ M = \pmatrix{1/2&0&0\\0&1/2&0\\0&0&2}, \quad b = \pmatrix{1\\0\\0}. $$ The sequence will converge to the limit $x = 2b$ if and only if the last entry of $x_0$ is $0$.