Theorem 1: Given vector b and invertible matrix A, which can be splitted as A = S + (A − S), where S is nonsingular, The iteration scheme \[ \mathbf{x}^{(k+1)} = \mathbf{x}^{(k)} + \left( - \mathbf{S}^{-1} \mathbf{A} \right) \mathbf{x}^{(k)} + \mathbf{S}^{-1}\mathbf{b} , \qquad k=0,1,2,\ldots , \] converges, that is x(k) → x for any starting vector x(0) if and only if \[ \rho \left( \mathbf{I} - \mathbf{S}^{-1} \mathbf{A} \right) < 1, \] where ρ(X) is the largest modulus of the eigenvalues of the matrix X, This number ρ(X) is called the spectral radius of matrix X.

See Forsythe, G.E. and Wasow, W.R., Finite Difference Methods for Partial Differential Equations. New York: John Wiley & Sons, Inc., 1960.

 

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