Gauss-Seidel Method

Updated 01 Oct 2026

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What it is
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  iteratively solves a linear system using the newest available component values
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Formula
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  $x_i^{(k+1)}=[b_i-\sum_{j<i}a_{ij}x_j^{(k+1)}-\sum_{j>i}a_{ij}x_j^{(k)}]/a_{ii}$
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Convergence
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  with $A=D+L+U$ , $T_{GS}=-(D+L)^{-1}U$ .
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  convergence from every starting vector requires $\rho(T_{GS})<1$ .
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  strict row diagonal dominance or symmetric positive definiteness are sufficient conditions.
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Related
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  Jacobi Method
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  Successive Over-Relaxation

Linked references 5