Stein-Rosenberg theorem

The Stein-Rosenberg theorem, proved in 1948, states that under certain premises, the Jacobi method and the Gauss-Seidel method are either both convergent, or both divergent. If they are convergent, then the Gauss-Seidel is asymptotically faster than the Jacobi method.

Statement
Let $$A=(a_{ij})\in\mathbb{R}^{n\times n}$$. Let $$\rho(X)$$ be the spectral radius of a matrix $$X$$. Let $$T_J=D^{-1}(L+U)$$ and $$T_1=(D-L)^{-1}U$$ be the matrix splitting for the Jacobi method and the Gauss-Seidel method respectively.

Theorem: If $$a_{ij}\le 0$$ for $$i\ne j$$ and $$a_{ii} > 0$$ for $$i=1,\ldots,n$$. Then, one and only one of the following mutually exclusive relations is valid:
 * 1) $$\rho(T_J) = \rho(T_1) = 0$$.
 * 2) $$0 < \rho(T_1) < \rho(T_J) < 1$$.
 * 3) $$1=\rho(T_J)=\rho(T_1)$$.
 * 4) $$1 < \rho(T_J) < \rho(T_1)$$.

Proof and applications
The proof uses the Perron-Frobenius theorem for non-negative matrices. Its proof can be found in Richard S. Varga's 1962 book Matrix Iterative Analysis.

In the words of Richard Varga: the Stein-Rosenberg theorem gives us our first comparison theorem for two different iterative methods. Interpreted in a more practical way, not only is the point Gauss-Seidel iterative method computationally more convenient to use (because of storage requirements) than the point Jacobi iterative matrix, but it is also asymptotically faster when the Jacobi matrix $$T_J$$ is non-negative

Employing more hypotheses, on the matrix $$A$$, one can even give quantitative results. For example, under certain conditions one can state that the Gauss-Seidel method is twice as fast as the Jacobi iteration.