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Convergence

The convergence properties of the Gauss–Seidel method are dependent on the matrix A {\displaystyle \mathbf {A} }. Namely, the procedure is known to converge if either:

The Gauss–Seidel method sometimes converges even if these conditions are not satisfied.

Golub and Van Loan give a theorem for an algorithm that splits A {\displaystyle \mathbf {A} } into two parts. Suppose A = M − N {\displaystyle \mathbf {A} =\mathbf {M} -\mathbf {N} } is nonsingular. Let r = ρ (M − 1 N) {\displaystyle r=\rho (\mathbf {M} ^{-1}\mathbf {N})} be the spectral radius of M − 1 N {\displaystyle \mathbf {M} ^{-1}\mathbf {N} }. Then the iterates x (k) {\displaystyle \mathbf {x} ^{(k)}} defined by M x (k + 1) = N x (k) + b {\displaystyle \mathbf {M} \mathbf {x} ^{(k+1)}=\mathbf {N} \mathbf {x} ^{(k)}+\mathbf {b} } converge to x = A − 1 b {\displaystyle \mathbf {x} =\mathbf {A} ^{-1}\mathbf {b} } for any starting vector x (0) {\displaystyle \mathbf {x} ^{(0)}} if M {\displaystyle \mathbf {M} } is nonsingular and r < 1 {\displaystyle r<1}.