What are the neccesary and sufficient conditions for the Gau

What are the neccesary and sufficient conditions for the Gauss-Seidel method to Converge?

Solution

The sufficient condition for convergence is if the matrix A is strictly or irreducibly diagonally dominant.

But for the convergence of Gauss-Seidel method (GSM) for a square system of linear algebraic equations, through the Generalized Line Criterion (GLC), (based on the absolute value of the matrix coefficients), following condition holds:

If A has a sign structure such that in each line, the diagonal element sign is opposite to the sign of all other elements, then GLC turns out to be a necessary and sufficient condition for GSM convergence.