# Solving linear system and obtaining operator norm

I need to solve a linear system of the form $$(\mathrm{Id} + \mathbf{J})\mathbf{x} = \mathbf{b}$$ for $$\mathbf{x}$$ and I also need to compute the operator norm of $$\mathbf{J}$$ (i.e. the largest singular value). The matrix $$\mathbf{J}$$ has no special structure other than being real-valued; the vector $$\mathbf{b}$$ is also real-valued.

I'm currently doing this in two steps, treating the two problems as essentially independent. I'm wondering what is the best numerical method for this problem.