5.3 Convergence
The GaussSeidel method was demonstrated in Sec. 5.2 using a sample problem, Eq. (5.2 ), that converged to within 0.2% accuracy in 9 solution sweeps. This section discusses the criteria for convergence of a solution.
The easiest explanation of convergence considers the error for each equation , deﬁned in Sec. 5.2 . Substituting in each term in Eq. (5.3 ), e.g. in Eq. (5.3a ), gives

(5.5) 

(5.6) 

(5.7) 
The solution begins with an initial error of , and . After one sweep the error is , and .
The error is quickly distributed evenly, such that and are almost identical at sweep 2. The errors continue to reduce since Eq. (5.7 ) and Eq. (5.8 ) guarantee that no error is greater than the average of the other errors.
Condition for convergence
The behaviour of this problem indicates a convergence condition for the GaussSeidel method: the magnitude of the diagonal coeﬃcient in each matrix row must be greater than or equal to the sum of the magnitudes of the other coeﬃcients in the row; in one row at least, the “greater than” condition must hold.
This is known as diagonal dominance, which is a suﬃcient condition for convergence, described mathematically as

(5.9) 