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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.

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